Escape learning is learning to perform a behavior because it terminates an aversive event and it reflects the power of negative reinforcement.
On the other hand, avoidance learning depends on two types of conditioning. Through classical conditioning an event comes to signal an aversive condition. An example of this is avoiding riding with drunk drivers because drunk people are known to have a high accident rate. A lot of avoidance behaviors are maladaptive though, and occur in response to phobias. Avoiding phobia triggering situations prevents the person from suffering the perveived dreaded consequences. The avoidence behavior, however, is negatively reinforced and therefore strengthened through operant conditioning. These type of behaviors are extremely hard to get rid of because people never give themselves a chance to learn that the dreaded consequences probably won't occur, or that they are greatly exaggerated.