Wedge anchor and drop-in anchor are both heavy-duty concrete anchors, but their structure and installation method are totally different.
Wedge anchor is through type anchor. You need to drill through the fixture plate, install the whole bolt assembly, then tighten nut from outside. It is suitable for surface mounted brackets and heavy load fixing.
Drop-in anchor is internal thread anchor, pre-set into concrete hole. After installation, only internal thread remains. You can screw bolt into it later. It works well when you need flush surface and hidden connection, like ceiling fixing.
Wedge anchor cannot be removed after full tightening; drop-in anchor allows bolt removal while anchor stays inside concrete.