Technically, a thief has the broad meaning of someone who takes property from someone--often in the absence of the owner. A robber is someone who confronts someone directly, in person, and takes something from them by the threat of violence.

Bandit and robber are the same. They steal things from people under threat of bodily harm. Bandits and robbits are obvious because they confront their victim face-to-face. A thief is stealthy and steals things from places. If he steals something from a person, he does it secretly by pickpocketing the person without that person's knowledge. The thief does not like to be noticed. He does things quietly and unobtrusively.