The word "hate" has somehow become an umbrella term for racism, sexism, homophobia, xenophobia, and any type of wrong attitude toward a group of people. It's very handy to replace a really long list with a four-letter word, like how I halve the time it takes to make a grocery list by just writing "food."
But just like some of the things I buy at the supermarket are not food by any definition of the word (Swiffers, toilet paper, Hot Pockets), a lot of these wrong attitudes don't involve the emotion we normally think of as "hate." "Hate" sounds like neo-Nazis stockpiling ammo and posting all-caps rants about blacks and Jews on, uh, Pinterest, or wherever neo-Nazis hang out (not a neo-Nazi expert).
I was joking, but, uh ...
Meanwhile, "hate groups" are groups whose sole purpose is to be against some other group, like they are so obsessed with being against gays or something that they have formed a club to spend all their spare time figuring out how to stop them.
This level of anger and obsession is what most people think of when you talk about "hate," and most racists or sexists or whatever do not feel that emotion. I think it's pretty obvious that most of the racism in this country is carried out by people who don't think they're racist, which is why it was so bizarre that Brad Paisley's "Accidental Racist" song implies that unintentional racism is some unusual kind of racism people need to be made aware of, instead of pretty much being the official racism of America.