Forget is a general word, forget about is more specific.

• “I forgot my keys.“: Person forgot their keys, as in it was left somewhere, and the person doesn’t have it.
• “I want to forget about my keys.“: Person would like to “un”know about its keys, or wants to bury the memories of it.
• “I forgot that face.“: Person doesn’t remember how someone’s face looked like.
• “I forgot about that face.“: Non-direct way of saying one doesn’t remember all the facial features – but not the entire face.