A criminal is anyone convicted of a crime, of breaking the law. We usually reserve this very negative appellation for those committing serious offences.
A bandit is a very old-fashioned term not in common use any more when referring to a type of criminal. It was a robber or outlaw belonging to a gang and typically operating in an isolated or lawless area.
The term was used as slang by the RAF during the WWII to refer to enemy planes: "Bandits at 2 o'clock!" and may still be used in one or more of the armed services still - I don't know.