The three terms get tangled together on American menus, so here is the short version.
Mochi is the dough itself. Glutinous rice, steamed and worked until it turns smooth and elastic. On its own it is barely sweet. In Japan it shows up everywhere - grilled, in soup on New Year's Day, dusted with kinako. When a menu says "mochi" it usually means small pieces of this dough, plain or lightly flavored. Ours is a cream mochi, made with milk and butter in the dough, which is part of why it stays so soft.
Daifuku is what happens when you wrap that dough around a filling. The classic filling is anko, sweet red bean paste. Ichigo daifuku - a whole fresh strawberry tucked in with the anko - showed up in Japan in the early 1980s, and several shops still claim they invented it. We make ours by hand: fresh strawberry, sweet anko, soft mochi, assembled the morning you eat it.
Mochi ice cream is the newest of the three. Same structure as daifuku, ice cream inside. Cold enough to hold its shape, soft enough to give when you bite. It is also the easiest one to hand somebody who has never had Japanese dessert.
If you are stuck at the counter, go ichigo daifuku for something soft and fruit-forward, or mochi ice cream for the cold chew.