Comparison of adverbs

Either one is correct, although, of course, the two sentences have different meanings.

She danced most elegantly.--means that her dancing was very elegant.

She danced the most elegantly.--means that her dancing was more elegant than that of anyone else.

If you're looking for distinctions about comparative adverbs, you'll need to come up with different sentences. She danced elegantly means simply that. She danced more elegantly is the comparative, meaning her dancing was of a higher degree of elegance than unnamed others. She danced the most elegantly means that her dancing was more elegant than everybody else.

This elegant dancing BS is a rather lame way to explore comparative adverbs, try something else.

You'll have to excuse Setanta, Igor-Lili. He get this way when he knows that he is getting out of his depth.