I'll be interested to see other with more experience in LED comments, but here are my thoughts:
It will come down to the amp draw of the device in question. Anytime you run below the a/h rating of the battery you get exponentially more run-time. So non-linear, right?
And I think most LED's are the inverse against voltage. That is, the more voltage you give them the more amperage they will also absorb..
No, if you're talking a 2600 lion @ 3.6V vs. a 2200 Nimh @ say 2.4 (2xaa), then the lion will be brighter but probably less long lived (although close).
Now run that as a 3xaa, so they're both @ 3.6v and the 2600 will be lots longer. More then the math would show, and that's over this non-linear reaction batteries of of life vs amp draw.
On the other hand, if you're only talking about lion to lion, and wondering about life vs ah rating, then I see no way the higher capacity battery could not live longer. The voltage of the battery and the draw of the associated driver should, in most cases, gate how much amperage is pulled from the battery.
I think... but I'd sure want to have that checked before you believed me..
