The difference is that we as a first person plural is generally used for a more discrete group of people, but still from the perspective of a single narrator. Fourth person we is generally used for a collective of people with a shared perspective; there is no single narrator that is separate from the collective group, the entire group is there narrator. Fourth person is a concept that has only recently begun to be recognized as a distinct point of view.
If you believe in string perspective there are infinite narrators narrating each other's narrations and we have only just started to make words for the infinities.
Doesn't exist. We/us is first person plural. Some languages have a little complexity here (e.g. Tagalog has "kami" which means "we without you" and "tayo" which means "we with you," but they're both still first person plurals).
Some linguists disagree and have recently begun accepting the existence of a fourth person point of view. Languages evolve, and I guess we'll just have to see if it catches on and becomes more widely accepted in the future.