Cleaning is accomplished with solvent, light oils, and elbow grease, but I think you're actually asking about the post-clean lubing.
Grease on sliding surfaces, oil where I want it to seep in. I apply grease with my finger after removing old grease with rags/swabs.
Grease won't seep out of your weapon / into your holster the way oil can, obviously range toys can be lubed with whatever the heck you like.
As far as why, my personal reliability testing has revealed that a linted-up dirty pistol runs a bit smoother with grease, so worst-case reliability is the driving factor. You'll have to do your own testing, I'm lucky that my carry guns all either benefit or aren't affected by the use of grease on rails, feed boss, hammer face, etc. I've let my two most carried guns get good and cold a few times, the grease doesn't seem to be degraded, probably because I use very little. Also consider that a carried gun will be close to body temperature until drawn, and then it will be heated from within or returned to the cozy holster ... not a lot of times it will get seriously cold for carry.