What's the difference between "menage" and "family"?

There certainly is a difference... ménage would sound and feel antiquated or (overly) formal to most people.

Furthermore, there is a subtle difference in meaning. A family implies that the members are a group of people related by blood, and they can still be a family if they don't share a house. A ménage refers rather to a household, which in many cases would be a family, but it doesn't have to be the same. A priest living with his housekeeper (once a common arrangement!) could be considered a ménage of two, but it's not his family. Similarly, a couple of students that share accommodations could be considered a ménage, but not a family.

And finally, ménage is used in the expression ménage à trois, which can lead to unwanted connotations when you mean a household!