There are at least 3½ reasons to do so, and only the first is commonly mentioned:
On #3, this is enough of a gray area that even police are encouraged to get civil permits.
And on #4, sure, you might think you'd know if you were a prohibited person, but on the other hand, you might have completely forgot about that little disqualifying incident 20 years ago.
Having the permit also qualifies as a second form of photo ID in situations where that might arise, but in general you want to minimize the number of people who know that you carry. Rely on your pilot's license instead. ☺
Showing my concealed carry permit is the last thing I’d use as a second form of identity.
Imagine some left wing lunatic seeing the permit, panicking and yelling at the top of her lungs: “GUN! GUN!, He’s got a GUN!”.