They are similar concepts, but not the same. For example, A child may run away from home, if he is angry at his family. A man may escape a prison.

Escape has more of the idea that someone is trying to prevent you leaving. So people 'escape' from prison because they are locked up. If someone's family is trying to stop them running away then you can use escape. She ran away from home three times before she finally escaped. Can also use these in a more abstract way. He always runs away from problems - he doesn't want to deal with them. When she is feeling stressed she escapes into her deams. She avoids her problems for a while by thinking about good things.