"Damper" is a term that describes the primary function of a shock or strut. Both are intended to control spring oscillation. They do this by damping the spring's natural tendency to oscillate by using hydraulic fluid. An un-damped spring will vibrate for quite some time after its initial compression or extension. Dampers (struts or shocks) usually reduce the duration of the spring's oscillation to one complete cycle.
Shocks usually have few other functions than to act as a damper. They are sometimes also used to limit suspension droop travel, and to limit suspension jounce (bump) travel.
Struts have a very important secondary function. They are an integral part of the suspension. While a shock can be removed with only minor consequences, when a strut is removed the suspension will collapse, unable to hold the car off the ground once the important link provided by the strut is removed.
1979 to present Mustangs all were designed with a strut front suspension and shocks in the rear suspension.