When you use "as" in such a context, it means "because this is what it is". "Like" means "in a manner similar to".
So if you say, "We have to face the problem AS a family", that would mean that we are a family and therefore we should face the problem in a certain way. If you say, "We have the face the problem LIKE a family", that means that you are not literally a family, but that your approach to the problem should be similar to the approach that a family would take. I would not expect a husband to say to his wife, "We should do this like a family", because they ARE a family, they are not LIKE one. A politician might say that the nation should "face a problem LIKE a family", because the nation is not a family, but in this case he believes it should act like one.
That said, "like" is sometimes used loosely in more the meaning of "as", especially to emphasize the appropriateness of some action. People sometimes say you should "act like X" when you are, in fact, X, if in the past you have not been acting like X. Someone who believes his family has been behaving inappropriately might say, "It's time we started acting like a family again". Or if someone is behaving immaturely, you might say, "Come on, act like a man" when he is, in fact, a man. Etc.