The changes are minor. Mostly some words are changed from the UK to the US versions to better suit the American vocabulary. Here's some examples I can think of offhand:
Philosopher's stone instead of sorcerer's stone; boot instead of trunk (for a car), lie-in instead of nap, chat up instead of hit on, lift instead of elevator, some of the food names have been changed.
And so on.
A lot of the British terms are in the US editions:
Scarpers, mental (as in, "He's completely mental!"), knackered, Blimey, trainers (sneakers or tennis shoes).
Obviously, there are spelling changes between the UK/US versions:
Humour/humor; colour/color; defence/defense; immunise/immunize; cosy/cozy.
Some of the Potter words have different capitalizations - Dementor versus dementor; House-elf versus house-elf; Centaur versus centaur; etc. The UK versions have single dialogue quotation marks, 'Like so.' The US versions use double quotation marks, "Like so."
I think it's worth it to have both sets of books. They're a British series and are a bit more authentic in British English.
Q: Does it bother you that in America they changed the names of your books?
JKR: They changed the first title, but with my consent to be honest. I wish I hadn't agreed now but it was my first book, and I was so grateful that anyone was publishing me I wanted to keep them happy. . . HP LEXICON
I'm glad I have both versions. There are absolutely no differences in plot between the two versions. If you have the money and the desire to read the UK versions, I'd say go for it.