A cloak is a full garment worn (usually outdoors) over all other clothes for warmth and protection. It is usually cut as a circle or partial circle, and is usually sleeveless but open at the front to give the hands and arms freedom. Cloaks are rarely worn in western cultures today; they have mostly been superseded by overcoats.
Robes are much more various. In mediaeval usage the word signified a sleeved garment with an ankle- or floor-length skirt, worn for non-strenuous activity; descendants of that sort of robe are usually rich, ornamented mantles worn over court dress on ceremonial occasions.
In more ordinary contexts a robe is a garment worn in very casual situations over swimsuits or pyjamas or nothing; it may be of very heavy or very flimsy fabric, depending on what activities are contemplated.
In Rowling's universe the word appeals to the mediaeval tradition, as a garment appropriate to sorcerers, but sober rather than flamboyant, like the gowns worn by present-day barristers, judges and scholars.