I have two Aimpoint H-1s and two EOTech XPS 2-0s. They are both first-rate military grade red dot sights. They are pretty different in design philosophy.
The EOTech has the signature circle-dot reticle for which EOTech is known. This is a 65 MOA outer circle and a 1 MOA center dot. This approach has the advantage of perhaps a bit quicker acquisition. Put another way, if you pick your rifle up and want to quickly acquire a target, the EOTech circle-dot reticle probably is easier to quickly acquire. The downside to it is that some people say that the outer circle portion of the reticle can be distracting, and some would say that they prefer a simple red dot for a reticle.
Here is the reticle on the EOTech XPS2-0
The Aimpoint simple red dot reticle is small, bright, and lends itself well to precision shooting.
The Aimpoint hardware (i.e. the H-1) is somewhat smaller than the EOTech XPS series. Both are small enough for a medium sized pistol or revolver, however, and both are ideal for about any rifle.
The EOTech XPS series are open holographic design while the Aimpoint is a tube design.
The Aimpoint battery life is nothing short of sensational. They use patented circuitry that allows an Aimpoint to go for literally thousands of hours between batteries. The EOTech XPS series is rated for about 500 hours, give or take depending on intensity. The EOTech will shut itself off after either 4 or 8 hours. For non-combat situations the EOTech goes for a long time between batteries.
Hope this helped. I own both and like both.