What can cause a wet patch to appear on an outside wall around an external tap?
My first guess is that the threads where the valve is screwed onto the pipe have started to leak. Even with the water turned off, there will almost certainly be water in the line that would leak. With the water turned off, you can try unscrewing the valve from the wall (as long as the plumber didn't solder the threads, it should come off). If the valve is relatively new and the threads are in good condition, then you can clean the threads, apply plenty of pipe dope to the threads, and screw the valve back on.

Note that if you don't see any mineral buildup around the pipe or threads, then the problem may be inside the wall and require removing drywall to locate the problem. If that's the case, I wouldn't delay in opening things up, since a small sign of water outside could be concealing lots of water inside.