This was a big debate at the time of their relases (1999-2000). Total classics, unanimously two of the most important multiplayer shooters ever released, and are both very, very similar. Which one is your favourite, and why? If you have not played both games for a reasonable amount of time online and off, then I ask you not to vote as you will probably give a biased opinion.
Quake 3 is my favourite, mainly because it is more simple, and therefore a purer and more entertaining experience. UT's alternate fire modes, while they work well and make things interesting, I feel that choosing between modes slowed down the action. While one could say that alterenate fire gives UT the more tactical edge, it doesn't necessarily IMO make it more entertaining. This same point goes for the translocator, which I rarely use. I enjoy using the movement system in Quake III much more, as it is faster and more flexible, and this is complemented by bunny hopping, strafe jumping and rocket jumping, which is used to greater effect in Q3. I cannot say which game had better level design. UT has a very interesting variety of maps - temples, spaceships, factories et al. I did feel that Q3 maps are slightly more practical though, and adapt to the movement system better than UT (i.e. more flow). Because of the technological differences, Quake does looks better and has more detail, but I see little point in taking that into account as it does not affect the gameplay.