In the HP series, there are many instances of students being bullied - the purposes of which mostly fall into demonstrating the consequences of bullying or the pointlessness and toxicity of abusing those weaker than us. Though there are many characters in the series who have, at some point, been abused, ignored, neglected or assaulted by people in their peer group or those who have responsibility over them, I've chosen to focus on the two that seem to create the greatest contrast to me: Neville Longbottom and Severus Snape. My main argument will be that there are vast differences between their circumstances and the characters themselves, and thus it can be argued that JKR has written, with strong intent, that Neville as a victim of bullying deserves more sympathy and is far more respectable than Snape; in the context of the books, it is Neville who is shown in a far better light in regards to bullying than Snape is.
By doing this, JKR, of course, operates under several assumptions:
a) that every fan will like Neville more than they like Snape (which, much to (I imagine) JKR's dismay, is incorrect)
b) that every person can tell the difference between the circumstances behind Neville and Snape. Given that this is the fandom that spawned 'fans' which publically wished she'd died in childbirth for not writing in their chosen pairing, I highly doubt that a lot of people know how to operate microwaves or open doors, and that the 8-14 year old market that the books were originally focused on is probably a lot smarter than we of the adult fangroups.
But nonetheless, I will continue.
Firstly, an introduction to the characters: Neville Longbottom. The relevant facts, as we know them:
a) He is an only child who lives with his grandmother, Augusta (GoF, p523)
b) His natural powers, when he first arrives at Hogwarts, are either far weaker than average or more unrefined (PS, 93)
c) He is bullied by several of the Slytherins, including Draco Malfoy (PS, p110; p160; p162; OotP, p321) and the head of the house, Severus Snape (PoA, p96; p107)
d) He has lost both parents to the cruciatus curse at the hands of Barty Crouch, Jr (GoF, p523) - who demonstrated the curse to him under the guise of Alastor Moody (GoF, p189)
e) Throughout his life, his grandmother and family have constantly made comments about his lack of skill in magic (PS, p93; OotP, p454; HBP, p165)
So, from these basic facts, a few things can be assumed:
*Despite his pureblood status, he's considerably less adept magically than his peers when he first arrives at Hogwarts, leaving him as more vulnerable than others in his grade, especially in comparison to teachers;
*Due to constant disparaging comments about his intelligence and ability by family, teachers and peers, would have a low sense of self-esteem in this regard;
*From his being bullied, and from losing his parents, he has a strong sense of right and wrong; he will fight back against perceived injustices; this is also evident from the instances where he is strong-willed and independent in defiance to others' actions - for instance, telling Malfoy that he is worth 'twelve of him' (PS, p162), trying to prevent the trio from going out to destroy Lord Voldemort (PS, p198), and insisting that he too fight in the Department of Mysteries (OotP, p671).
Next, we have the character of Severus Snape. Likewise, a few relevant canonical facts:
a) He attended school around the time of Harry Potter's father, James, and his friends Sirius Black, Remus Lupin and Peter Pettigrew (PoA, p261)
b) Little is known about his upbringing, save that his mother, Eileen Prince, was a pureblood, and his father, Tobias Snape, was a muggle, (HBP, p594) which in turn makes him a half-blood; there may be some evidence that he was unhappy in his upbringing after watching his mother and father fighting (OotP, p521)
c) There was at least once instance of him being bullied in school, as proven by canon, where James Potter and Sirius Black attacked him for what appears to be no good reason (OotP, p569)
d) He was academically advanced for his age, as proven by Sirius Black (GoF, p460) and by the fact that he invented curses such as levicorpus and sectumsempra (HBP, p563)
e) He was renowned at school for both being adept at dark magic (GoF, p460) and for associating with people who were widely known to become Death Eaters after they left school (GoF, p461)
f) He instantly loathed Harry Potter before he even met him (PS, p101), despite the fact that a) he'd never known his father to have heard anything about Snape in the first place, and b) he was an orphaned ten year old boy who had no opinions about anybody in the wizarding world save for the man who murdered his parents - Lord Voldemort. In addition, he has made unkind comments about a teenage girl's appearance when she had done nothing to slight him (GoF, p263), and persistently bullies Neville Longbottom (ibid)
g) He was visibly and loudly furious when his school-enemy, Sirius Black, escaped, thereby avoiding an unfairly delivered Dementor's kiss (PoA, p306)
h) He has been described as being quite physically unfavourable (OotP, p522), and was likely not popular with teh ladies (OotP, p569)
i) In response to someone trying to defend him from bullying, he hurled what is considered to be a nasty racial epithet at Lily Evans (OotP, p571)
Likewise, similar assumptions can be made about Snape's personality and his circumstances:
*He likely was unhappy as a child - both due to the scenes in 'Snape's Worst Memory' and Harry penetrating his mind in an Occlumency class;
*He likely held beliefs considered racist in his world, and associated (and later joined) with the Death Eaters;
*He was quite intellectually skilled as a child, and later used his skills in his career as an educator, where he held considerable power over people under his care; due to his treatment of children under his care, and his lacking of skills as a teacher, it's fair to assume that he is very much a bully in his adult life;
*He has not recovered from his experiences as a child, nor has he forgiven any slights, whereas his enemies have openly expressed regret and remorse for their treatment of him (OotP, p590) - he constantly longs for vengeance and takes offence at even the slightest of insults.
Thus, we see a contrast immediately: Neville Longbottom, who has arguably and canonically, had an actually more traumatic upbringing than Snape's (of which this is still speculatory), and who is also more vulnerable due to his age and lack of skill.
As victim's of bullying, Snape and Neville both react in different fashions. Snape, when learning of Professor Lupin's particular teaching methods in relation to the Boggart, takes his frustration out on his students, in particular Neville - despite the fact that Neville had no idea what Lupin would do with his particular phobia:
"Snape didn't seem to find it funny. His eyes flashed menacingly at the very mention of Professor Lupin's name, and he was bullying Neville worse than ever." (PoA, p107)
In addition, his childhood vendetta against Remus Lupin was so long-standing that he placed him in a situation where he had no choice but to resign from his job (PoA, p308); despite the fact that Remus Lupin did not actively bully Snape, it did not prevent him from taking out his anger many years after the fact in a completely vindictive fashion - knowing that Lupin likely would not have been able to find any other paid employment to support himself, and that it would result in even more ostracism from his society and the possibility of even violence or criminal action being taken against him. On whole, Severus Snape's reaction to those who bullied him ranges from violence (OotP, p569), swearing (OotP, p571), and vindictive punishment (PoA, p308; p264; p107).
On the other hand, Neville tends to react with non-violent, positive self-assertion. In response to Draco's taunts, he tells him that he's worth twelve of him (PS, p162). When he accidentally eats one of the twins' joke store products and turns into a canary, much to the laughter of all those in the room (GoF, p320), he too joins in with the laughter - though people are laughing both at and with him, he too is able to have a sense of humour about the situation and understand that it wasn't a personal offence. However, on occasion and when extremely provoked, reacts angrily to genuine insults against him: for instance, where he lunges after Draco having heard him make comments about those with mental injury (OotP, p321), or as Bellatrix Lestrange taunts him about his parents in the Department of Mysteries (OotP, p706). To me, this would seem to indicate that JKR has written Neville to be someone who picks his battles, and though he understands that it is wrong, will only react with violence to something that is truly offensive to him; in contrast, JKR has written Snape as someone who will pettily react to even the slightest of insults: even if it comes from a child who doesn't know better or understand his history. Thus, JKR has demonstrated Neville to be more mature and of better nature than Snape in regards to bullying.
Secondly, the issue of abuse of privilege needs to be raised, given the vast disparity between the power, ability and privilege held by Snape in contrast to Neville. In canon, Snape is often shown to abuse the trust of the students he teaches: whether it be teaching incorrect facts in order to just slander a colleague (as evident in the kappa debate in PoA (p129) and Fantastical Beasts), or making cruel comments about an aspect of someone's appearance which he knows is quite a sore topic with the person (when he makes the all-time most bastardly comment "I see no difference" in response to Hermione's teeth being hit by a growing hex (GoF, p263), he is constantly either neglecting his duties as an impartial adult educator by picking favourites, having scapegoats and filling the children's heads with untruths and arguing with people about areas they are quite knowledgeable in. I consider there to be a huge difference between bullying amongst one's own peer group and age level and the abuse of power by an adult over a child, and though Snape was undoubtedly bullied as a child it doesn't excuse him abusing his powers as a teacher when he grows older.
On the other hand, Neville (who is often bullied by Snape) is someone who, until his lessons with the DA in OotP, is quite vulnerable in terms of his ability. For a large part of his schooling he is less-skilled than his peers, and due to his childhood and comments made about his intelligence he is also more emotionally fragile. Not once in the series is he seen to make so much as an impatient remark about anyone younger or weaker than him, and he is often seen to get along well with Ginny Weasley and Luna Lovegood, both of whom are a year below him. When he does react with violence or anger to bullying, it tends to be someone who is an equal to him (such as Draco Malfoy), or someone much more older and skilled than him (like Bellatrix Lestrange or the other Death Eaters). Thus, Neville is not a bully, as he does not physically, emotionally, intellectually, mentally or magically intimidate or assault anybody in the books who is weaker than him in any way. One then assumes that given these factors, it is indeed Snape who is a bully, as written by JK Rowling in canon - not someone like Neville.
Finally, the issue of provocation will be raised briefly. Though many bullying attacks and assault in the real world, just like in the wizarding world, is based on prejudice and often has no motive or provocation behind it, there are instances where people incite anger from other people or offend them in some way. Though James Potter did utter the legendarily cruel words "well... it's more the fact that he exists, if you know what I mean" (OotP, p570), there are several hints within the novels that he may, indeed, provoke some of the anger towards him. His 'mudblood' call against Lily Evans (OotP, p571) may indicate that he has been prejudiced against those because of their status of birth in the past, which might understandably raise the ire of some students regardless of their heritage - that he would also utter it at somebody who tried to help him might also give some insight into his civility to other students at the time. Severus Snape also did invent two highly dangerous and provocative spells (HBP, p563), and it is unlikely, given a) what we know of his temperament, and b) what we know of the peers he chose to spend time with at school, that he did not actually use these spells against students in the past. It's also probably worthwhile to add that while this provocation by no means excuses or condones the bullying against Snape, it does provide the context for the bullying and makes it a little more understandable. On the other hand, there is precisely no provocation for Neville's bullying: it is all based on picking on a much weaker student, and taunting him about his parent's job and downfall - neither of which he had anything to do with.
In conclusion, JKR has indicated precisely how much pity we need feel for Snape as a victim of bullying, in comparison to someone like Neville Longbottom. While I myself find Snape a fascinating character (rather like a meaner George Costanza), as both a victim of bulling and (to a far, far mild extent) a proponent of bullying*, I find it hard to sympathise with him due to the fact that as a thirty-something man, he has failed to mature or seek the necessary kinds of emotional healing required for maturation. Similarly, he refuses to bestow the same kind of pity that he demands upon people he either sees being bullied, or bullies himself, and it's for this reason that JKR presents him in a mostly unsympathetic light. While she does want us to understand that Snape is a fantastic example of the consequences of bullying for many people (as if real-world examples such as the Littleton, Colorado massacres didn't teach us as well), she also demonstrates that if any sympathy for victims of bullying is deserved, it is deserved by those who forgive, don't abuse their power, and have no prejudice against others for factors they cannot control.
*On a far more personal note, this is something for which I have apologised to each and every single person involved. Carrying on childhood spats and insults is something I barely have the energy for with people off-line, and I can hardly understand how someone would have the motivation or effort to maintain such grudges until they're in their mid-thirties.