I own one of each. Range/woods use only, my trips to the middle east don't involve carrying firearms.
Eotech: Great quality, reticle pattern (65MOA ring with 1 MOA dot) is great. I have a N-cell version, which means that it's nice and short.
Aimpoint: Sets the standard for reliability and battery life. I don't find that the single dot is quite as quick in a real close-range setting, but it's still very fast. I've actually got a 2MOA dot, 2X version in a "scout" type mount. It's all the scope that I'll need out to at least 300 meters.
Long story short...try to find a retailer that stocks them, and look through them. If you can't do that, be happy knowing that both companies make a solid product and you'll be getting a solid sight.