In German "blöd" means stupid and as in this excersise the adjective "stupid" was an optional answer I thought "Yeah cool, that one is easy: Who wants a stupid dog. Enter." Nanaaaaa. False friend.
And in Swedish, nasse is slang for either "pig" or "nazi". So Wer möchte einen nassen Hund? doesn't sound that good in reverse, either. :p
They're essentially synonymous. The only difference I can think of is that if something is only a little wet, slightly more than moist, you wouldn't perhaps use blöt. But that's really nitpicking. :)