In my mind, "inheritance" is almost exclusively used to deal with physical or monetary goods, features, or character traits; you can inherit your father's estate, you can inherit your mother's good looks, you can inherit your grandfather's business sense.
"Legacy" can be used in that same sense, but it can also have a much broader sense to refer to a situation that is set up for for the benefit of the descendents: "My father's legacy was the freewheeling, open, friendly town that bears his name: Fatherville." Or, "Our parents' legacy was not simply that they bequeathed us ten million dollars each, but that with that money in trust, we felt free to explore, experiment, and take risks that we would not have been able to, had we not had that safety net available."
(Also, as pointed out in some comments, "inheritance" is generally from the viewpoint of the receiver, while "legacy" is from the viewpoint of the giver: My inheritance was my father's legacy.)
When you say "mind" you are saying "be aware of" or "be careful with" not to be confused with "beware of".
They might overlap, depending on why you may want to be aware of something. You might need to "mind the dog" because he bites or you might need to "mind the dog" because you are about to absent-mindedly step on it's tail.
"Beware of your head" is strange sounding. "Mind your head" means to be careful with your head. "Beware of" in this context implies that your head is going to do something bad to you, whereas "mind your head" is meant to say be careful not to let something bad happen to your head.
Best Answer
"Beware" indicates danger, and "be aware" merely calls attention to something.
For example:
"Beware of the dog" - it might bite you.
"Be aware of the dog" - you might trip over it in the dark.