Membrane elements have three degree of freedoms (DOF) per node. Two DOFs are in-plane translations and one DOF corresponds to the out of plane rotation. Think of balloons for example.
Shell elements have six DOF per node. All three in-plane translations and rotation along all axes. This is one of the reasons we commonly prefer them to continuum (solid) elements.
The shell element captures the bending stress as well as the membrane stress while the membrane element ignores the bending stress. Membrane elements are used to represent only in plane stiffness of members. It acts like the bed sheet on your bed. The bed sheet does not have any kind of bending stiffness and so it bends without any resistance. So, use this kind of element when you do not want slabs or any other plates to resist bending I mean out of plane bending.
Shell elements on the other hand are more realistic for any structural wall or a slab or basement walls or even thick pipes. They have in plane as well as out of plane stiffness.
Practical Aspects of Finite Element Simulation: A Study Guide, Altair University.
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