I think it would be a relationship that involved lots of forced smiles, awkward silences and thoughts of "Oh. It's you..."
But I don't think they'd hate each other. They just wouldn't be able to spend too much time together. I don't think Aurora would like Ariel's disregard for authority, her impulsiveness or her lack of family values (I'm not hating, I just mean in comparison to Aurora). While Ariel would get slightly bored of Aurora over time.
I think they'd get along because they dream of meeting a guy and falling in love, but I also think Ariel would quite quickly get sick of Aurora's pleasant-ness, passive-ness and lack of curiosity. Based on this, I think they'd be friends that sometimes get along, but more often than not, they don't
I agree with Mongoose. It's the shy, calm older girl who likes quiet time and the hyper younger girl who craves excitement. it would be hard for them to find anything to do together. as Mongoose said there'd mostly be forced smiles LOL. Ariel would like Aurora's romance but dislike how she goes to the castle instead of running off with Phillip and think she obeys the rules too much. In turn, Aurora would find Ariel to be a little girl who needs to grow up and mind her elders.