I would say that "hold-up" is a subset of "robbery". A "hold-up" generally takes place on official premises, such as a bank or jewelry shop; or it involves vehicles in transit between such places; it also implies the use of firearms or at least weapons of some kind.
"Robbery" on the other hand can also include e.g. a confrontation in a dark alley, where someone relieves you of your valuables with the mere threat of a beating up.
Not a professional ESL teacher.