Yes, smaller diameter filaments can be used in printers which support higher diameter filaments and not vice versa. This value is not to be confused with extruder diameter which is considerably smaller than the filament diameter.
The amount of filament pushed into the extruder is calculated according to the filament diameter value given to the printer software. The supplied filament is then heated and extruded. The amount of plastic that flows out of the extruder is thus governed by the software that is supplying the filament to the extruder.
Secondly, the diameter value is important because, the heat has to reach the center of the filament for it melt effectively. In the case of thinner filaments, it is obvious that there would be no problem at all. Hence, usage of specified filament is not a problem, but any filament thicker than that would lead to clogging issues.