There are two basic types of mechanical wedge grips.  Either the grip bodies move and the wedges remain stationary while being tightened on the specimen, or the wedges move as they tighten on the specimen and the bodies of the grips remain stationary.  In Wyoming Test Fixtures, Inc. terminology, the former are termed “standard grips” and the latter are termed "simple grips".

Because the wedges in the “standard” wedge grips do not move axially relative to the specimen during clamping, the grips do not have the potential of putting the tensile specimen in compression, possibly damaging it as the grips are tightened. This is obviously a more desirable condition, in general.  However, the Simple Tensile Wedge Grips are shorter in overall length (and thus better able to fit into confined spaces), and are slightly easier to fabricate and thus are slightly less expensive.