It was always a different feel. PC shooters were at their peak and were moving at the speed of light and then Halo came out and brought everything back to a snail's pace. It's just that people who played Halo when it came out didn't know any better because it was their first shooter.
Every shooter since then has just been... gradually trying to recover from what Halo did to the genre. They're still not back to where they were.
Yes and no.
The pace of Halo was in part a result of trying to make the genre work on consoles with a slower input method.
That said, the solutions to that problem led to mechanics that pushed the genre in a more interesting direction than bunny-hopping around maps like jack-rabbits on crack.
Halo was the first FPS that demanded you think out your approach tactically, prioritize enemies appropriately, bring the right tools for the job, use your environment to your advantage... and wasn't completely punishing if you failed any step of the way. The recharging shields mechanic encouraged players to try things, just so long as they were smart enough to retreat and reengage when it was clear their current strategy wasnt working.
By comparison, tactical twitch shooters on PC were offering an extreme response to the extreme speed of the Quakes and Unreal Tournaments, where they would bury players for even the smallest mistake as a way to try to break them of habits formed since Doom.
Halo’s systems were intuitive, the inherent balance of choices self-evident. Combined with great feeling shooting (a combination of visuals, sound, reticule friction, etc), Halo is still largely the game most console shooters are chasing. In place of actually reaching or surpassing its raw satisfaction, the genre has turned to filling exp bars and other carrots on sticks.