"... not A but B" vs. "... B but not A"

The point is not urgency but importance.

The point is importance but not urgency.

Normally, we use the first construction. It feels that the second sentence has a somewhat different meaning. Is this indeed so, or do they in fact have same meaning?

The difference is in the rhetorical usage. The reason we usually see the first version is because it dismantles whatever previous notions the audience had. The speaker is telling the audience that it's not the thing they initially thought, it is something else. Saying "not urgency" in the second case makes no sense, because you have already established what the point is, so we can assume that nothing else is the point.