Membrane and Plates have basic differences in structural behavior. Membrane sustain transverse loading using inplane stresses. While plates sustain loading using bending stresses. 1-D analogous examples are difference between a thread/cable and beam. If the bending stiffness of the plate is reduced to zero, it will act like a membrane with due consideration of boundary conditions .
A plate (solid) can bend. Remember, a plate has only translation degrees of freedom and no rotational degrees of freedom. Bending is not rotation, understanding this difference is important. Plates sustain out of plane loading by bending stresses.
A membrane cannot bend. It can resist forces by in plane tension (transferring load to the supports via. tension). Membranes sustain out of plane loading by in plane stresses.
A shell is a combination of a plate and a membrane (that is it will have bending as well as in plane stiffness).
plane stress elements are for in plane forces. plate elements are for out of plane forces. When there are both in plane and out of plane forces, then typically a shell element is used which is a plane stress and a plate element combined together. Even though the word "shell" in structural mechanics refers to membrane stresses in FEA shell elements normally can take both in plane and out of plane forces.
PLANE182 is used to model 2-D solid structures. It can be used as either a plane element (plane stress, plane strain or generalized plane strain) or an axisymmetric element with or without torsion.
SHELL181 is suitable for analyzing thin to moderately-thick shell structures. It is a four-node element with six degrees of freedom at each node: translations in the x, y, and z directions, and rotations about the x, y, and z-axes.