If Buffy had lived in Sunnydale with her father, and if their relationship had stayed the way that it was during the summer between Seasons 1 and 2, no, I don't think that Giles and Buffy's relationship would have been as strong, unless Giles had been a woman, so that SHE could have replaced the MOTHER figure that Buffy had lost.
And, yes, I do think that there would have been some significant changes. To begin with, Hank would never have let Buffy go out with anyone as old as Angel APPEARED to be. No responsible father would let his 16/17 year old daughter go out with a 25 year old man, no matter how charming he was, and once Hank found out the truth about him, Angel wouldn't be allowed within half a mile of Buffy, and they would have had to come up with a totally different reason to have Angel lose his soul, assuming that he lasted all the way to Season 2 at all. (Don't get fired up, Bangels, the same thing would have gone for Spike, as well, though, of course, that relationship didn't really get going strong until Season 6, at which time whichever parent would have been dead) Parker would have been thrown out of the house before the first date, and Hank would have had Riley in abject terror within 2 minutes of the time that he showed up to pick Buffy up for their first date (no dorm room living for Hank's little girl!) but after a couple of weeks, they would have been going to ball games and car shows together.
The first time that Dawn shouted "Get out. Get out! GET OUT!!!" Hank would have explained to his pseudo daughter that HE was the one who made the house payments there, and any "Getting out" that was done would be her disagreeable little a**, off to boarding school in Montana.
Those are just a few differences that I could come up with on the spur of the moment. I'm sure that someone like Joss could have come up with a whole lot more.
