Each had standout maps that made you want to own both, such as Q3A's Longest Yard and UT's Facing Worlds. I ended up playing more hours of UT because I had slow internet and its bots were better.
Point being: Q3A was great, but in 1999 it became clear that id Software had lost their head start over other FPS developers. They were still elite, but in the early-mid 90s they were alone at the top.
https://www.gamespot.com/reviews/quake-iii-arena-review/1900...
https://www.gamespot.com/reviews/unreal-tournament-review/19...