First you shouldn't confuse demoiselle with mademoiselle, or dame and madame. The latter words are never written a splitted way (
ma demoiselle, ma dame.)
The word demoiselle is outdated and almost only used in the fixed expression demoiselle d'honneur.
The word dame (lady) is a polite alternative to femme (woman). Especially when the person is listening to you, it would be considered rude to name her as femme instead of dame.
We would rarely say Il y a une femme qui vous demande and never dis bonjour à la femme !, that rather be il y a une dame qui vous demande and dis bonjour à la dame !.
On the other hand, mademoiselle and madame are titles used to prefix names (Miss, Mrs), or to call someone (Miss, Madam, Ma'am). The obsolecence of mademoiselle for any non child female person has already been discussed. See Do French use Madame to describe non-married women? Comment s'adresser à une femme dont l'état civil est inconnu ?