ETA: For your two choices, I would probably go with the Aimpoint just because of the battery life. I think I have left every battery powered optic I own on at one point.
The current version F of the Eotech:
1,000 continuous hours (lithium) at nominal setting 12;
600 continuous with AA alkaline
I don't know. With the Eotech and normal shooting, the batteries will probably go stale before they die from use.
If the Aimpoint batty lasts 50,000 hours (5.7 years) if left on, then I guess the batteries will last their shelf life....unless you leave it on for 5.7 years?
It's kind of a moot point for either, since the batteries are cheap, readily available, and last a long, long time. I would not use that as a criteria for purchase. The average person may use it 2hrs. a week, at best. That works out to 100 hours a year of use.
I would get what is easier for me to aim through.