Just a general question. Do Lithium batteries make an LED shine brighter than Alkaline batteries? Or are they recommended just because they keep supplying power more consistently than Alkalines?
The answer is probably YES
But it really depends what your question is exactly.
By "Lithium" do you mean the AA or AAA silver Energizer Lithium batteries? Or are you meaning something like a CR123a Lithium or maybe even a Li-ion rechargeable battery?
There are several things to consider. Firstly the battery power and performance. And then how the light is designed to operate.
1. If we look at the battery first.
A normal Alkaline battery like a Duracell or lots of other makes. These have 1.5v (v for volts) when resting. This is true for AAA, AA, C or D cell sizes. The 'bigger' the battery, the longer it lasts, i.e. it has a higher capacity or mAh.
The trouble with alkaline batteries is, firstly they leak and ruin things, so should generally be avoided. But when you apply "load" to the battery, i.e. having a torch on, especially a high output mode. The voltage sags (gets lower). And they don't generally allow much current.
This means that very soon after turning a light on, the battery will sag and cause a performance drop. They also get considerably worse as they reduce capacity. This means for any 'high output' device, they perform really badly, such as torches, digital cameras, etc.
A Lithium battery like an Energizer Lithium is a direct replacement. They also make 1.5v but they key differences are, they don't leak. They don't mind the cold, they have a really long shelf life. But most importantly they allow more current and sag a lot less. This means you will generally get much better performance from these. More current in some lights could mean more output (depends on the light). And more stable voltage should give more consistent light for longer. So better runtimes. Sadly this isn't always true as some torch makers program their lights to not take advantage of this. But a lot do.
A good NiMH rechargeable AA/AAA is only 1.2v but exhibits potentially even better consistency, although ultimately less capacity. But will generally easily output perform an alkaline battery.
Something like a CR123a has 3.0v, so a lot more punch. Plus the good consistent current and little voltage sag. Although there isn't a direct alkaline alternative. So few lights will run on this or alkaline.
Li-ion is rechargeable. These have 4.2v, so considerably more punch. And again don't sag like alkalines. So these will generally offer a lot more light, but it does depend. The higher voltage can also fry torches not designed for 4.2v, so you need to know if your torch is compatible. However there are some torches that allow you to run 3 x AA or 1 x 26550 Li-ion, the latter is massively better.
2. Torches generally have 'drivers' that control the modes and output of a light. These can be boost drivers where the battery voltage is lower than needed, so the driver will need to pull current from the battery to make the LED work and feed it the correct voltage. There are buck drivers where the voltage is higher than needed and the driver will lower it down, but not need as much current normally. Efficiencies of these can vary wildly. Or you can get linear drivers which make about the right voltage from the battery for the LED. These will often allow the LED to be 'direct driven' from the battery. Which means it is all down to how the battery performs on how much light is produced.
Drivers can also be programmed to limit current, dim the light or step down. This can be after a period of time, thermal limits or even battery sag levels. So the answer isn't 100% unanimous for all situations. for example, Maglite (and I think Led Lenser) often program their lights to dim, to maximise performance on alkaline batteries. Putting a 1.5v Lithium AA in them will not improve performance, because the driver program will negate the benefits of the battery. But you may still experience better total run times and no leaks. So they are still worth it.