If you are referring to membrane elements (finite element analysis), then standard membrane elements have only in plane translational degrees of freedom and thus account for in plane/membrane loads/stresses only (not bending).
Shell elements in your case plate elements, since the slab is very likely flat (shell refers very often to curved thin structure), have both translational and rotational degrees of freedom and thus account for bending as well. In essence the plate/shell elements are a combination of an in plane membrane element and an element accounting for bending.
Membrane elements are used to model thin structures (membranes, fabrics,..) where there are membrane (in plane stresses) stresses only. In order to build up some bending capacity in a membrane we need to add a tension stress, hence in FEA, a nonlinear geometric analysis needs to be run, which can do that, and where the bending stiffness builts up as we get tension in the membrane element (hence why these elements are used with a geometric nonlinear analysis, since this is a nonlinear effect). Of course if the membrane element is stiff, not like a fabric, one might still be able to run it linearly (no large deflections), but then again only in-plane membrane stresses are captured.
Shell/plate elements have bending stiffness. Plates/shells can be used to model slabs, etc and other thick structures that can resist bending loads (a non-tensed membrane cannot take bending/transverse loads). If the plate structure is also thick thus transverse shear is important there is a thick plate theory known as Mindlin, while if the plate is thin, then simple Kirchhoff theory (no transverse shear) is enough.
Very often FEA software will let one choose which one to use.
So it depends what type of structure (thin/thick) and loads (in plane or transverse) that you have when it comes to deciding which of the two finite elements to use.