I don't remember any specfic reasons why they dislike each other, but Ursula was born 1 minute earlier and did everything earlier so was deemed the better of the two to family and friends.
In episode 1.16 - The One with Two Parts (Part 1), the following conversation takes place
RACHEL: Um, Pheebs, so, you guys just don't get along?
PHOEBE: It's mostly just dumb sister stuff, you know, I mean, like, everyone always thought of her as the pretty one, you know. Oh, oh, she was the first one to start walking, even though I did it, later that same day. But, to my parents, by then it was like "yeah, right, well what else is new"?
Also, the fact that Ursula doesn't care about Phoebe's feelings, including not showing her the suicide note that their mother left (Episode 4.01 - The one with the Jellyfish)
Ursula: No, I umm, I read about her in Mom’s suicide note.
Phoebe: There-there was a suicide note?! (Ursula nods ‘Yes.’) Well, do you still have it?
Ursula: (disgusted) Hang on. (She goes into her apartment and slams the door in Phoebe’s face.)
Phoebe: I can’t believe you didn’t tell me there was a suicide note!
I'm sure there are more references etc. but this should help.