The difference between the two is bright nickel is a thin plating of nickel deposited over standard carbon steel by an electro-plating process.
Nickel can be rather easily damaged by scratching or chipping, and cannot be repaired by anything other than stripping the finish off and completely re-doing it.
When cleaning a nickel plated gun you have to be careful with bore solvents around it since many solvents can infiltrate under the plating and cause it to flake off.
Stainless steel is a steel and it's stainless all the way through. If it's scratched the metal can be simply re-polished.
Stainless is not easily damaged by any bore solvent unless it's a really aggressive solvent and you let it soak on the metal for hours.