It's too difficult to point out all the differences. But I have some notes here:
Affair is used when someone takes it for a short time, maybe long time but not whole life.
Mission is the same, however there may be missions for whole life and a mission usually lies in a chain of various missions to complete a ready prepaired plan of one person or one group. All of us also has our own affairs but only some people have their own missions.
Business is used as both short time work and whole life work.
It mainly means job or career
Work can be used generally in many contexts. It can be trifle or important things that someone have to or must to or should or want to do.
That's only my own idea. Maybe not true at all. Sometimes expression is more difficult than doing what in your mind
Thanks