Deformable Remote Force (RBE3 element in a linear analysis) distributes the force without adding stiffness to the structure. If you perform the same calculation to determine how the RBE3 will distribute the force to each node and apply the nodal forces yourself, you will get a very similar result.
Forces applied to nodes do have a say in the displacements in the model: F = K.X, double the force, double the displacement. Forces don't have a say in the stiffness of the structure, K = F/X which is the same as 2F/2X.
Rigid Remote Force (CERIG element in a linear analysis) rewrites the stiffness matrix, giving you a stiffer structure than what you had before you added the Rigid Remote Force. When using this, think about adding very stiff elements between the scoped faces/edges.
Forces do not become part of the global stiffness matrix [K], they are in the system of equations that are solved [K]{X} = {F}.