I don't hate the U. I guess I pity them, really. Such a sad, shallow existence they must lead. But I've been a lone Cougar in RES, and I've seen a lone Ute at LES, and the difference in how each loner was treated is night and day.
I think it really started to get nasty in 1977, we sent Wilson back into the game late so he could set an NCAA record, and the coach at the time went nuts:
http://en.wikipedia.org/wiki/Holy_War_(Utah_vs._BYU" onclick="window.open(this.href);return false;)#The_LaVell_Edwards_era
Of course, McBride upped the ante, mostly be actually being competive. Meyer upped it again by pure douchebaggery - although I get that second hand, since I was in Uruguay for the duration of his short stay here in Utah.
More than the coaches, though, I think the hate has really increased due to demographic changes in Utah and the U. SLC is becoming a lot less LDS, and so is the student body at Utah. And my impression is that there aren't a whole lot of people on the fence when it comes to the church in SLC - you love it, or you hate it, and all things associated with it. I think that's the main fuel on the rivalry hate fire. In contrast, I don't get a lot of hate from Aggies I know. It's fun, good-natured rivalry, even now that they've been competitive. Probably because Logan is WAY different than SLC.