I'll separate this one into a few categories. The first is construction. We'll skip the obvious visual differences, in practice it makes no difference unless you are carrying your rifle in a fashion show, all models from both manufacturers have adequate lens protection and robust cases. The weight depends on the model, at the featherweight end of the spectrum the EOTech XPS weighs 8 ounces, while an Aimpoint T1 with a tall mount weighs about 6 ounces.
The adjustments are where the sights take different paths. On the Aimpoint the windage and elevation adjustments are under aluminum caps, while on the EOTech they are out in the open on the side of the unit. While I wouldn't be concerned about the adjustments being knocked off zero on the EOTech, there's still no reason not to have them protected, and for that reason it's an obvious downside. The brightness adjustment on Aimpoints is a twist knob, while the EOTech uses push buttons. This comes down to personal preference for the most part, but I prefer the Aimpoint's system, as it is far less likely to be bumped accidentally, and is a simpler design thus less likely to fail.
The EOTech has it's mounting system built into the unit. It provides an absolute cowitness on an AR with standard sights. The upside to this is cost, you don't need to spend an extra $100-$200 on a mount like you do with an Aimpoint. There are several downsides though, all coming down to limited options. There are a variety of mounts for Aimpoints, including 1:30 type mounts for a cant type setup, various heights so you can use the standard lower 1/3 cowitness for an AR, absolute cowitness, or low mounts for handguns and AKs. The bottom line here is cost vs. adaptability, so I won't say one is better than the other.
The last and most hotly contested difference is the reticle. The two sights project the reticle differently, the Aimpoint being a reflex sight and the EOTech being holographic. There's a good explanation of the two designs here. For practical use though there isn't much of a difference between the mechanisms, both are nearly parallax free and will hold up to serious abuse. The reticles on Aimpoints are a 2 or 4MOA dot, while the EOTech is a 1 moa dot surrounded by a 65MOA ring. Many people will say one is faster, one is more accurate, but as I will explain later in the article neither is true. The reticle choice is 100% personal preference. Using both and deciding which suits you is the only way to know which will perform better for your needs.
This is a tough one to determine. Search for any information on it and you will come up with a few thousand forum posts of "I have x and have had 0 problems, thus it's invincible and will never fail." Of course there is no way to determine whether this poster subjects it to a thousand rounds a week or has it mounted on a nerf gun. You can refer to more reputable shooters/competition community, but then you have the problem of endorsements or people not wanting to burn bridges with manufacturers or suppliers. The most popular is looking at various militaries and LE agencies, and the classic "HERE'S A PICTURE OF A NAVY SEAL USING ONE, IT'S THE BEST." I can also show you a picture of a Chauchat being fielded though, remember the military has cost, availability, and political issues that are weighted a lot more than actual function or value. That being said, L3 Communications (manufacturer of the EOTech) have lost their government contract, leaving Aimpoint as the only player.
EOTechs do have a major issue with the battery contacts. The sights can fail under recoil, specifically but not limited to the 553. Aside from that both the Aimpoints and EOTechs have a low rate of failure and excellent customer service. Based on track record and design though Aimpoint edges out EOTech in the ruggedness department, and most experts will agree on this.
There are a few things I hear quite a bit when this debate comes up, and I assume the people that are bringing it up either haven't used the sights in question, have only used them in video games, or are using their red dots incorrectly. The first is field of view, the argument being it's larger on the EOTech because of the larger window. This is complete nonsense - since there is no magnification and little to no parallax, you will use either sight with both eyes open and the reticle will be superimposed on your target. You cannot see the enclosure of the sight, or even the tint of the glass. It would make no difference whether the window was half the size of a T1 or as large as the windshield on your Daewoo Lanos, as long as you can see the reticle the FOV is the same. In fact, you can even have the front of the sight completely covered and you will see little difference on your target.
The second is accuracy, the thought being since the EOTech is a 1MOA dot and the Aimpoint is 4MOA the EOTech must be 4 times as accurate. This, again, is nonsense, as your rifle does not know the size of your dot it will shoot exactly the same no matter what size dot you are running. As far as practical accuracy, for any RDS the dot size is far down the list of variables. On a well defined target a decent shooter will not find a difference in accuracy between the two. Personally I find the Aimpoint to be easier to shoot precisely, since the dot is not opaque, on a low power setting it's easy to place your target in the center of the larger dot. Here's a nice demonstration of the capability of a 4moa dot for precision shooting.
I saved this for last as I believe this is by far the most important difference between the two sights. EOTechs models offer a few hundred hours of battery life while Aimpoints most popular models offer 50,000 hours+. The EOTechs also automatically turn off after a set amount of hours. A red dot sight that's not on is as effective as a toilet paper tube taped to your rifle. The ability to be left on all the time and not be turned off by anything but the user or a nuclear bomb is a 100% requirement. There's no reason to add in the extra step of turning something on when you may be under stress, being shot at, on fire, etc., when there is an option to avoid it. For this reason the choice is clear for any rifle or handgun you may need to rely on someday: Aimpoint.