Hmm. It's certainly a thing to chew on- but I think it does sort of have some elements of sexism meshed in. Not necessarily in that people hate Umbridge more than they hate Snape, due to the fact that she's a woman, but because of her position and how her character gets 'punished' and how Snape gets punished.
Snape terrorizes his students and is a quite prejudiced character, but people hate Umbridge because she's a character that we have all encountered in our lives.
She's an agent of the meddling government who intrudes and is flat out aggressive towards those who don't fall into the boxes she sets for people. But it's interesting because I think Alan Rickman does have some part in why people like Snape.
It's kind of odd, and I never really got into that kind of thing, but it's sort of a fangirl effect? He's tall, dark, and british. (Handsome is, of course, subjective). In the books he's described as oily and hawkish and all around abrasive. And anyone who has only read the books, will probably agree that he's generally unpleasant and a necessary* evil. Everyone encounters that one teacher that despises you and your guts and will purposefully (for whatever reason or bias) try to fail you in any way they can.
But in the movies, Snape is given this odd angle- because the movies weren't released in parallel with the books, fans of the books weren't going in blind. So they gave him a little more leeway, and cut out the bits where he was truly horrid because going in, we already /knew/ he was horrid and could fill in the blanks. They were arguably reasonable cuts, because translating the book word-for-word would lead to many hours long sagas and that wouldn't bode well for many. (Though I am quite upset about the bits that were cut. Peeves was one of my favourite elements of the books.)
So you've got a watered down Snape, made up of the following elements.
~ Not very much screentime, and thus his cruelty is limited to maybe a few lines per movie. But, for good reason- He's not the lead! Harry and his friends are the focus of these films!
~ Played by Alan Rickman, who is not exactly awful to look at, and who is quite good at what he does. And since he was acting with knowledge of the character's then unreleased history, he added the sympathetic edge to the character that may have made him more appealing.
Whereas Umbridge was relatively untouched in the cutting room, because she was in two of the books. And she was a main antagonist, so her actions got more spotlight. In addition, her appearance in the movies were more accurate to the books.
She was basically designed to invoke that feeling of frustration when figures of authority step in and make your life hell- both as a teacher and as a student.
While Book!Snape was also designed along these lines, Movie!Snape... not so much.
But while maybe there isn't as much sexism in this as it is hokey screenwriting, there are some elements that are fundamentally different in how Umbridge and Snape are treated.
For example, the centaur scene, which highly implies some very very questionable things- I won't spell them out, but look up the mythology of Centaurs and their typical role in ancient myth.
This 'punishment' for Umbridge, was laid down by Hermione. And later it's played for laughs in the hospital when Ron torments Umbridge by clicking his tongue like hooves. Yes, Umbridge is a rampant racist, and she's dreadful to non-humans, and to non-pureblood wizards, it's important to remember that SNAPE WAS TOO.
Snape's racism is played down considerably in the movies, and I think that that's used to sort of mentally Woobify him on the part of fangirls, because without looking at all of his actions in context (Hate crimes, calling close friends by racial slurs, harassing and mentally and verbally abusing students, attacking the child of your percieved 'true love' years after her death simply for having the gall of looking like his parents, unhealthy relationships and power dynamics), you can forget that Snape is actually a pretty awful person.
Not to mention Snape's punishment for all of these actions (listed above) are... nothing. Basically nothing. Even in the books! Harassing, emotionally and verbally abusing students? A slap on the wrist. Calling Lily a slur? He loses her friendship... and then proceeds to never ever grow up or learn from his actions. He dies! But his death is very much played down, especially in the movies, which had a very obvious fondness for the Snape/Lily ship.
In the end, Harry even names his child "Albus Severus"! (Which, I'm sorry, I still think is a stupid name.)
Snape gets a backstory, and rationalization for his actions:
~ Bullied as a child,
~ Abusive parents,
~ Difficulty socializing,
~ Had an actual attempt on his life as a teenager by the Marauders, and was told to bottle it up and never talk about it by an authority figure he trusted.
But he ALSO (and it's important to remember this is largely glossed over in the movies):
~ Joined a group of racists who were actively campaigning to kill people exactly like his so-called best friend.
~ Has killed people.
~ CALLED HIS BEST FRIEND A SLUR, when being what she was (Muggleborn) placed her life in actual literal imminent danger.
~ Harassed a student to the point where in a world where your parents have been tortured to the point of never recognizing who you are, and Voldemort is Real, and it was an agent of Voldemort who did that /to your parents/, your greatest fear is a teacher.
~ Was willing to sacrifice James and Harry's lives so that he could be 'rewarded' with Lily, /regardless of her feelings/.
Yes, these things are awful, but intent does not, and cannot excuse actions, you know what I mean?
Plus, we know nothing about Umbridge or her motivations. She works for Fudge at the ministry. She's racist. She's fanatical in setting rules, and she can be very short sighted.
I'm not saying that that's bad, and sometimes people are just awful regardless of upbringing, but I mean, look at the differences in how these characters are handled.
So while I don't think it's so much sexism on the fandom's part that Umbridge is detested more, I think it's sexism on the part of the screenwriters on their portrayal of these two very similar characters.
*Necessary as in no matter what you're going to always face these people in your life. Not that they're built into the system, but that sometimes you literally have no recourse but to take it and once you're free, /leave/ and never go back.
Man this was way longer than I meant to make it. Whoops!
-Bryn.