There seems to be quiet a lot of misinformation on this thread.
The primary difference between a mouse and an analogue stick is that a mouse allows for muscle memory. While the importance of this can't be understated in competitive play, I'd go as far as saying that the majority of people who play with a mouse and keyboard aren't actually good enough to take advantage of it. If you play with an excessively high sensitivity or acceleration turned on, which a shockingly high number of players seem to be doing, you aren't actually that much better off than if you have a controller. Furthermore, if you play stupid, it doesn't matter if you have the best aim in the world. You're going to get blindsided and shot in the back. Case and point: every person killed in this video was probably using a mouse and keyboard, and at least half of the shots could have been easily done with a controller.
While raw adjustments are harder to do, good players using a controller will make smaller adjustments by moving. Incidentally many higher level PC players will do something very similar, simply because moving makes you harder to hit and therefore gives you more time to aim. The skills that actually help you win in a competitive setting, such as positioning, pre-aiming, pre-firing, good timing, and good communication are very much applicable with a controller. Flick shots are not relied on, because they are way too random to be counted on, especially when the enemy is doing everything they can to make sure that they aren't going to have to rely on a flick shot.
Now, keep in mind that certain games will benefit from a keyboard and mouse more than others. Games like quake and tribes tend to rely more on raw adjustments. I will also admit that, despite all of what I have just said, I will still take a mouse over a controller any day, simply because saving myself with a flick shot feels unbelievably awesome.
So, to recap and give a TL;DR version:
The mouse is better, but unless you're a competitive level player, it doesn't make as much of a difference as PC players would like to think.