When to Use Lightweight Animator vs Unity's Animator ?

for my game's UI ?

That's where the Lightweight Animator Controller realy shines. Unless you want to bother with curmbersome Triggers, Bools and such, you can probably use the Lightweight Animator, and different layers, to achieve what you have in mind, with Hovers, Click, Selected states and such. Aslo, if you have an animated prefab inside a list, with a lot of instances, your game's performances will suffer way less.

for world environements, or small interactable items ?

You can probably use the Lightweight Animator Controller. Especially if you start instantiating a lot of these. Performances are way less impacted by the Lightweight Animator Controller, than Unity's own.

for my main character's animations ?

You will probably prefer to use Unity's Animator. It costs more performance, but if you want humanoid animations, blend trees, maybe Layer Masks, you are way better off with it.