You would typically only use "face" when the subject is a human or animal. The idea is that when you "face" something you stand up to an obstacle without turning away to run from it. Since only living creatures can do that, typically it should not be used on an inanimate object or a concept. So the first phrase is perfect, but the second one is a bit off.

In my opinion, "face with" indicates the situation of that you have to find a way around. Face sounds like "subject" is a recipient, for example: Clinton said the U.S , Afghanistan and Pakistan face a "common enemy" with extremists. I am not a native speaker, this is what I am thinking.