Well I watched Angel first and will always prefer it. But I do see them as two different shows so I dont think you have to watch one to watch the other. But I think you miss out if you don't watch both.
That's exactly my view as well.
I knew literally nothing about Buffy when I saw the original trailers for ANGEL leading up to its premiere on October 5, 1999. I tuned in not even knowing Angel was a vampire. Once that drunk glided out of the bar, double-staked the vampires, revealed himself to be one, told the girls off and took off down that infamous alley, leading into the awesome opening credits, I was completely, utterly hooked.
The next week revealed even more about this guy, and just how much we were alike in so many ways. It was like looking at me on screen, but infinitely better-looking than I could ever hope to be with the best of plastic surgeons.
I didn't see Buffy at all except for the crossovers, because I deliberately avoided paying any attention to it. I tuned it out while waiting for this to come on at 9, back on those WB's New Tuesday nights. I didn't know anything about it, so I figured why bother if I'm going to be completely lost?
Finally caught up with Buffy when it hit the rerun schedule in September of 2001, a couple of weeks before season 3 of ANGEL premiered and even then - I was only interested so I could see Angel, Cordelia, Wesley and Darla, and their history and events I missed out on originally. Eventually the show itself did grow on me, but fan talk turned me off of it quite a bit because of the constant Buffy / Angel series comparisons and debates about which was supposedly better, when to me they are (and still are) completely different shows that happen to be set in the same universe. I grew to dislike it out of spite to a large degree, but that's gone now. I don't compare them, because I CAN'T compare them. ANGEL remains more personal to me because of how much I relate to the man himself, but as series I love them each for very different reasons - appealing to different personal preferences and personality aspects. Comparing them would be like comparing children or pets. Can't do it, and it wouldn't even be fair to if I could.