What's the Benefit of Using Layer Weight? I just don't see the difference using Facing and Fresnel, or Layer Weight as a whole in my node editor.

Is it a big difference using it, is it useful for distant objects?
There are many big differences, among which are that:


Fresnel depends on the face normals of your mesh - it will not work with inconsistent or incorrect normals;
The Layer Weight node allows for a smooth blend between the two;


Depending on what you wish to achieve, it may be better to use a specific one, but generally, you'll want to use the Fresnel node, as it's more physically accurate.