Copper-lined stainless steel vs pure copper skillet
As a general rule, a solid copper pan and a "copper-lined" (either plated, or possibly a thicker layer bonded in some way) are likely to be quite different in terms of heat distribution, since stainless steel is a very poor conductor of heat and copper is a very good one.

So, when heated "unevenly" a copper pan of a given thickness will conduct heat to even out  the heat the food sees, while a stainless steel pan will be much more prone to hot and cold spots. A blend or sandwich of the two will be somewhere in between, and exactly where in between is it will depend on how much of the sandwich is copper, .vs. how much is stainless steel. Copper is (at present, and for quite some time) far more expensive than stainless steel, so there's a strong economic incentive for it to be thin, at the manufacturing point.

A stainless pan that was lined with the same thickness (or nearly the same thickness) of copper as the copper pan alone was would be pretty similar to the copper pan, but I don't know if anyone makes something like that. The whole pan would be thicker, in that case.