What is the difference between "legacy" and "inheritance"?
Solution 1:
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.)
Solution 2:
Legacy, as it relates to inheritance, will be of broader or deeper scope. When inheritance refers to one generation's gifts to another, a legacy would refer to three or four generations.
Both terms can be used to denote objects other than money:
I have inherited my father's eyes.
Drinking is the family legacy.
Either an inheritance or legacy can be positive or negative. Legacy does carry a slightly more fatalistic connotation with it: The legacy will live on through the children.
Hellion's observation of legacy's use with regards to a specific individual is also completely valid. The word can be used to describe a legend or long-standing effect caused by one person ("this is his legacy") or even a family, nation or race ("this is their legacy").
Solution 3:
'Legacy' specifically has the additional related meaning of 'a person admitted to a university because of the attendance of a parent at the same school' often with the implication that it was done in disregarding merit of the applicant. 'Inheritance' has no such meaning.