In a real world situation, the lower amp controller will tend to help you get a better average wh/mi number.
Lots of varaibles with such a vague question, but here is what I see on my dirt bike. 2812 9 c motor. I ride a 72v 20 amp controller in summer, in the hope that I will be less likely to melt down the motor when climbing steep hills. So with that controller, the max watts I see no matter what is about 1500w. The max grade I can climb is about 15% and riding as hard as I can, I see wh/mi in the 40's. Max speed is 25 mph with the slow winding.
Once weather is reliably below 80F, then I put a 40 amp contoller on it. Speed goes up only 1 mph to 26. Watts max seen on the CA is about 2500, but mostly it's still pretty hard to find hills steep enough to draw 2000w. I can now climb steeper hills, so I do, and this costs me in wh/mi. I tend to see wh/mi in the 50's.
So what it amounts to is kind of like this. If you can, you will, and that can cost you wh/mi. Even that tiny 1 mph speed increase still costs you something. But on the street on the flat, you don't see a ton of measurable difference. On the hill, you see a huge difference, because you will tend to climb the hills faster, and use more power.
In another example, with a 9c 2807 motor, going from 40 amps to 20 amps simply costs you speed if the ride is on the flat. The 40 amp will draw 2500w, and go nearly 40 mph. The 20 amp won't have the watts to go 40. 1500w will only get you to 35 mph. So in this case, the lower amp controller simply cannot reach the same speed. But compare the two controllers at 30 mph, and both will draw about the same wattage at that speed.
What to do in the real world? Well, I always prefer to solve any ebike problem by carrying more battery if that is possible. Unless you are going to melt your motor, solve your efficeincy worry by adding wh. Worth it for a more enjoyable ride. If you are carrying all you can, and just need to get spectacular wh/mi on a particular day, you simply do that with the throttle. Slow down to the watt number that gets you the whole distance. If it's a daily commute that is the problem, and slowing down is not an option, then carrying more battery is the mandatory solution.
Lastly, if your problem is poor efficiency climbing a very steep hill, like 8% or above, then the regular windings of hubmotors are a poor choice. At the same speed up 8% say 13 mph, a 2810 winding beats a 2807 because the 2807 can't tolerate such a slow speed at full throttle. The winding makes the difference climbing slow up hills. Not the controller.
