Paired together like that, in their meanings of "favorable or unfavorable attitudes toward a person," they're roughly synonyms.
Sometimes it's helpful to go back to root meanings. "Bias" means "tilt" or "slant." "Prejudice" means "judged in advance." On that basis, we might say that a prejudice is harder, firmer, and allows no exceptions.
"This boss has a bias against Martians (space aliens from the planet Mars). Non-Martians with 'good' performance get promotions, but only Martians with 'outstanding' performance get promotions."
An example of prejudice would be a boss who says "No, don't bring this candidate for an interview. This candidate is a Martian. All Martians are inherently lazy, stupid, and ugly. There's no need to me to see if this Martian is diligent, smart, and handsome. I already know the answer."