When something is done sadly, it happens in a downhearted, unhappy way. You can speak sadly, hang your head sadly, or even smile sadly to convey how melancholy you're truly feeling.
Sad means unhappy or sorrowful, so if the adverb sadly describes someone's actions, they're feeling down in the dumps. The Old English root of both words originally meant "full or sated with food," and later came to describe a heaviness that remains in the word sadly. It's also used to mean "unfortunately" or "it is a sad fact that," so you might say, "Sadly, chess club is canceled for the foreseeable future."