General sheaves allow constructions that are sick and wrong. For example, you can define a sheaf that geometrically consists of two lines, so that every single point of the two lines are identified, but the two lines are still distinct. The "concrete" condition prevents that kind of pathology.

I think the argument for diffeological spaces is just that it eliminates certain kinds of pathological constructions that are possible with general sheaves, without costing you anything. General sheaves allow constructions that geometrically are sick and wrong. For example, you can define a sheaf that geometrically consists of two lines, so that every single point of the two lines are identified, but the two lines are still distinct. The "concrete" condition prevents that kind of pathology.