I purchased the Eotech in July of 2010. In January 2011 I picked up the Aimpoint M4. After shooting both, the Aimpoint is my choice, and here is why. Aimpoint is pictured top, Eotech below.

DurabilityIf you know me, then you know that I value tough products. I'll gladly pay a little more to make sure that what I got can handle some abuse. As you can hopefully tell from the pictures the Eotech 512 does have a strong aircraft aluminum hood which covers the main body. However, the battery compartment in the front is NOT covered. Doesn't mean its fragile by any means, but it is something that I look at.
The Aimpoint M4 is completely encased in an aircraft aluminum casing. Everything except the brightness control dial is metal. There was a demonstration where a rep from aimpoint threw the optic across a parking lot, threw it back, mounted it up and it still retained zero. Thats worse than anything that I would do to my optic so I feel pretty secure. The Aimpoint feels absolutely rock solid.
Battery LifeThe Eotech 512, using good alkaline batteries gets about 1100 hours on 2 AA batteries. The Aimpoint M4 gets 80,000 with 1 AA battery. No contest aimpoint hands down. I can leave the sucker on for years and not have to think about it. I'll probably even change the battery every 1 year just to keep it fresh.
The aimpoint power system is so effective at drawing battery life. I took a battery from my remote control. This batter was "dead", the remote stopped working. Using that "dead" battery, I put it in the aimpoint and voila, it still worked. Why? Because the Aimpoint can use the tiniest amount of power to run. My remote didn't even register the battery as having any juice left, but the Aimpoint found some.
The Eotech 512 has an auto-shutoff feature. It automatically shuts off after 8 or 4 hours, depending on which button you use to turn it on. This is a negative for me, because honestly I'd like to control whether its on or off.
As mentioned, the aimpoint has so much battery life it doesn't need an auto shutoff.
Mount

I call mounting options a draw, because I have a LaRue Quick Detach for both my eotech and my aimpoint. I like LaRue because their mounts are solid, guaranteed to retain zero, and are very simple to use. The Quick Detach is a great feature of LaRue mounts, you can take optics on and off very quickly, while still maintaining a solid zero. Also, LaRue will sell you an optic at the standard price and include their mount instead of the standard QRP2 (in the case of the AP).
The Eotech mounts to the weapon system directly whereas the AP requires a mount. However since I use a LaRue on both of them the Eotech's benefit of integrated mount is lost on me. Again have to call this one a draw.
Controls

Here's where it starts to really swing in AP's favor. The Eotech 512 controls as you can see are two rubber buttons, one for lowering brightness one for increasing brightness. The controls on the eotech 512 are not that bad until: #1 you have to lower brightness quickly, #2 you have a Backup Iron Sight sitting right behind it. #3 in my case, the buttons sometimes get stuck, they're rubber and sometimes they deform a little bit and turn sideways slightly, causing them to stick.
The Aimpoint uses a control knob. Clock wise for brighter, counter for dimmer. There is an audible "click" everytime. The control knob as you can see is away from the iron sight/magnifier,thus you can easily reach the controls with one hand without taking moving away from fire control.
When you're adrenaline is pumping, or you're getting fatigued, your fine motor skills become somewhat lost. Pressing little buttons is fine for many folks, but I prefer simply turning a knob, or because the battery life is so good in the case of the Aimpoint M4, not having to hit the knob at all.
Field of ViewField of View. This is where the Eotech is supposed to shine. With its window like display, it offers a wider view of things. The AP, being a tube design, is supposedly not as good, because of"tunnel vision". Let me tell you, tunnel vision is entirely possible with an eotech as well.
If you're shooting close up, then you should have both eyes open. I found that using the AP, I had just as much view of multiple targets/surroundings as with the Eotech. When not looking through the tube, I look over the optic anyway, so I know how not to get sucked into the tube.
Reticle (refer to pics above)
The targeting recticle on the Aimpoint is a simple 2 MOA red dot. The Eotech 512 is a 1 MOA dot with a 65 MOA dot circle around it. For me the simple dot was much easier to keep track of and see. When the Eotech recticle would bounce during multiple snap shots, it would look like a huge blur and it got a bit distracting. The AP's dot, although it did bounce as well, was less distracting. Some people like the big targeting recticle of the Eotech, but for me I can do without it.
All of this is strictly my opinion, there are many people on both sides of the fence, and people who really like both. Eotech makes a great product and I am happy to recommend it for people who like the way it handles. I chose my AP (just like everything else) because it works better for me, and my shooting style/preferences.