Both of the sights are used by the US military. However, Aimpoint just secured a contract for the Comp M4s model for the official sight of the US military. I believe they ordered over half a million sights. The Aimpoints are very rugged, almost impossible to knock off of zero, very water resistant (if not proof), offer a single 2 or 4 MOA dot for viewing, and have an 8 year continuous battery life from a single AA battery. That's right, I said 8 year life and that's with the unit on. All Aimpoints are parallax-free and will work in all light conditions and all temperatures "from the sub-zero temperatures of the Arctic to the heat and humidity of the tropics". That's a quote from their catalog. You can also mount a 3X magnifier behind the Aimpoint to make it a 3X scope. Also, you can use a PVS-14 Night Vision Device behind the Aimpoint if you have access to such things.
The Eotech offers a 1MOA dot surrounded by a 65MOA circle. This offers a little better reticle to pick up on in super fast situations and it's a range finding reticle but I can't find a lot of info on how to find range with it. The Eotechs are paralx-free as well but the battery life is only around 1,000 hours for most models. The Eotechs with the 5-1-* designation like the 512 or the 516 are not night vision compatible. The Eotechs with the 5-5-* designation like the 557 or the 553 are. If you aren't going to use night vision with them then there isn't really a reason to spend the extra $100 on the unit. Otherwise, the only difference is really the batteries used in the Eotech. Many use 2 AA while others use the CR123 battery.
Some people who have "older" eyes don't like the Eotech reticle. They say that it is blurry. This is usually due to the fact that the user wears bifocals and has a hard time picking up on the target and the reticle at the same time. Both of these sights are meant to be used both eyes open. The human brain combines the images in both eyes to make one image. This is where these types of sights rule the market because you have such a wide field of view and are able to shoot at several close range targets in a short amount of time. This feature also helps to engage moving targets.
Let me know if this helps your decision.
I know this is a lot to take in, feel free to email me at firstname.lastname@example.org
if you have any questions as to how to go about ordering one or prices.