I've never owned an Aimpoint, but I've shot with them a couple of times. I have an Eotech mounted on my duty rifle, and I love it.
In my opinion, compared with the Aimpoint, I feel that the Eotech provides faster target acquisition, and keeps my field of view open better. These are important considerations when using a rifle in a life-and-death situation, but may be less important to the avid sportsman.
Similarly, Aimpoint is known for having a long battery life (insanely long), and this fact probably doesn't mean a darn thing to the average user. Even still, I have to respectfully disagree with Sentinel on the "very short" battery life in the Eotech. I have carried an Eotech on my work rifle for probably 5 years now, and it has probably had around three battery changes in that time. I'd say that I get at least a year out of a set of AA batteries, and those batteries (as you know) are really easy to find. The rifle isn't deployed daily in my work environment, but it certainly gets regular use. I also use my rifle at the range at least a dozen times per year. So, my battery use wouldn't be as heavy as a soldier who is using it constantly, but my gun certainly isn't a safe queen.
I don't leave the reticle on when I'm not using the rifle, and it takes just a one button push to light the reticle from the off position (in other words, it won't really slow you down when deploying the rifle... faster to turn on the Eotech than it is to sling the rifle or charge the weapon).
Anyway, both options are viable, and both the Eotech and Aimpoint have a loyal following. I'm happy with my Eotech, and would make the same choice if I was going to buy again today. Just my $0.02.