head to head
Neither is meaningfully harder - they pose close to the same test. The widest gap is execution, and it is a single point.

Execution · Pure Mechanics
The sequel that built console online multiplayer around ranked Battle Rifle duels.

Execution · Pure Mechanics
Pure aim and movement skill tested against adaptive human opponents competing for spawn-cycle dominance.
Halo 2 leans Execution (Pure Mechanics); Quake III Arena leans Execution (Pure Mechanics).
Quake III Arena leans a little harder on execution, though the gap is small.
Halo 2
BR precision and movement tricks online
Quake III Arena
Rail and rocket aim at speed is the entire game
Quake III Arena leans a little harder on reads, though the gap is small.
Halo 2
Ranked duels hinge on outplaying humans
Quake III Arena
Human opponents adapt their routes and timings in direct response
Same score on variance - which means the number cannot separate them. The difference is in what each one asks of you.
Halo 2
Matchmaking opponents and spawns vary
Quake III Arena
Spawn locations have mild randomness but are quickly mapped
Same score on optimization - which means the number cannot separate them. The difference is in what each one asks of you.
Halo 2
Map control, timers, and weapon spawns
Quake III Arena
Armor and weapon timing routes are learnable and highly leveraged
A matching score does not mean a matching test. Two games can both sit at 10 on execution and demand completely different things of you - which is why the reasoning above matters more than the number beside it.