My notebook happens to have a fan calibrated to turn on when the CPU gets "too hot" where the point is above of what is consumed by IE to show the pages with more videos, so I actually
hear the power consumption of Firefox and I simply don't hear it for the same page when opened in IE.
So for me is simply: open some page, hear the fan for as long as the page remains open. Do the same with IE, silence.
BTW I otherwise haven't IE used for daily browsing, but I did a lot of tests with the same pages to see that it's not an accident and that IE does show the same content, but with less power used.
For somebody with a different computer: find any web page with more media playing on the page (you should see the videos playing or pictures "animating"), open the task manager (this is a Windows example) then switch to the graph view and compare the browsers while looking at the graph view. The task manager area "under the curve" reflects the power used -- the more CPU used for a given amount of time -- the more power used. In my observations the area drawn by Firefox is more that twice as big as the one by IE.
That's what drains the battery faster: taking more CPU to keep the "animations" on the page. Moreover Firefox has some consumption created by some of its background tasks even when the pages aren't animated, but the animations are really wasting a lot of power.
From my look at the behavior, having some experience in evaluating the performance of the software, and spending some time trying to figure out what's happening there, my impression is that the source of that power drain is in the Firefox code which is not platform specific which explains why they have problems on every platforms with that.
I've concluded they simply never actively tried to minimize overall CPU and GPU use (both will use more power unless somebody measures and tries to improve the code): even when they did move some pieces of processing to the GPU, their motivation was speeding up what's visible, actively minimizing the power used was simply not the part of their process.
You can also search their Bugzilla to see how they handle the problems with the CPU, GPU use and the power drain. This can be a part of the workflow, measuring that at least for every release and acting to fix the issues, but I just don't see that anywhere.