Quake 3.
Unreal Tournament has a ton of features and a more compelling single-player. In addition, its larger variety of stuff and powerful enforcer make the game great for when you want to play a competitive game that's not going to completely alienate the less skilled players. I thought it was the better game for a long time.
However, Quake 3 is just so flawless that it's on a level all by itself. All of the weapons are perfectly tuned and have a distinct purpose with the maps to match. The real selling point, however, is the absolutely perfect physics and movement dynamics. Overall, the design is so laser focused that no other game can compete, not even UT99, itself one of the best games in the genre. After some games of Q3A its competitors feel downright sluggish.
Franchise-wise, it's no contest: Quake.
Quake 3 > Quake > UT99 > UT2K4 >> Quake 2 > UT2K3 > UT3 = Quake 4