Let's just pretend the E1B and LD01 both have 80 lumens out the front.
The E1B uses an optic that focuses most of the light into a tight hot spot, with a fairly dim spill.
The LD01 uses a small reflector which creates a larger and dimmer hot spot and brighter spill.
The LD01 may run at 80 lumens for the life of the battery and then shut off.
Looking at the run time graph above of the L1, the E1B may maintain 80 lumens for 1.3 hours, but then drop below 80 lumens and continue to run for another hour, or whatever, dimming the whole time.
So the E1B should run longer on it's CR123 than the LD01 runs on it's AAA when both are set to max.
Lately Fenix has been more accurate with their specs as I believe they now use an IS and measure out the front lumens.
When it comes to Surefire, I have an E1B that is probably around 80 lumens, but others have E1B's that maybe closer to 120 lumens. The E1B's MINIMUM out the front lumens should be 80, but SF often exceeds their specs. They may have released the first E1B's with XR-E P4's which produced 80 lumens out the front, but now they maybe using Q4's, Q5's, etc, and producing 100-120 lumens. I figure they wouldn't change their specs to reflect the brighter emitters because there is probably still old stock out there with the older emitters. Since Surefire doesn't specify what bin LED they use it can be luck of the draw, but you should get a light that performs at the advertised spec at the worst, although the tint maybe another story.
So the E1B will certainly look brighter as it has most of the light focused into that small hot spot, but that doesn't mean the LD01 isn't producing a similar amount of lumens. In most cases I would expect an E1B to have slightly more overal output than a LD01, unless you stick a 10440 3.7v AAA in the LD01 and then it's game over for the E1B, at least when it comes to lumens. The LD01 will be putting out around 200+ lumens, out the front, with a 10440, for a short while anyway.
The E1B would still probably have more throw though.