It may be useful to look at the words in terms of their etymological origins. Help is Anglo-Saxon, assist is French. Help is used in concrete situations, in general, in ordinary life (e.g. help me by holding a ladder). Assist is used in formal, academic, and institutional situations, as in surgery, where it often means just watching and very occasionally doing something useful, or international diplomacy, where it might mean any sort of request someone else finds useful, such as not voting (abstaining).