"Hence" means "from here". "Get thee hence!" is old English for "get out of here". "Henceforwards" is "from this time on".
"Thus" means "in this way". In older English, you might say "he opened the door thus", while demonstrating the action.
In logical senses, you can use them in similar contexts. "If x is true, then y is true. X is true. Hence, y is also true." "If x is true, then y is true. I have proved x is true. Thus, I have proved y to be true also."
Think of them as "based on this fact" and "in this way" and you can't go far wrong.
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.)
Best Answer
I don't think there is a difference in meaning, only in usage.
This blog post details it:
Examples: