What is the difference between lithium and alkaline batteries, and why would I use one over the other?

I presume you mean Lithium primary batteries; not rechargable Li-Ion batteries??

Lithium primary batteries have double the shelf life of alkalines and their voltage is more nearly constant. They are also lighter in weight, and can output more current, and have slightly higher energy storage per unit volume. So they are definitely better by every measure.

But Lithium primary batteries are lots more expensive than a good alkaline. So the right answer depends on what you are doing. If it is a flashlight, I'd stick with alkalines. More expensive regulated flashlights are available for alkalines that are as good as the Lithium ones & you'll save a bundle on battery costs.

If it is a camera, use NiMH rechargables (buy at Target). The high current requirement will kill the alkalines too fast.

Alkaline batteries are not rechargeable. Generally the lithium batteries are.

Lithium batteries last longer.