Easy Answer: The polished nickel finish is a shade or two darker than the polished chrome finish. Also, a chrome finish will be much more resistant to corrosion than nickel anything, polished or otherwise.
Slightly Harder but Interesting Answer: Hardware made with a polished nickel finish is made of the element Nickel and hardware made with a chrome finish is made from the element Chromium, or at least the outer layer is made of Chromium (chrome plated).
Both metals are easily polished but Nickel starts and ends a shade darker than Chrome.
One of the main differences between these two metals is that Chromium is extremely resistant to tarnishing or corrosion. It is the hardest metal on the Periodic Table and has an extremely high melting point. For this reason there are about a billion different things (not even counting hardware) that are chrome plated. This is done by electroplating a thin layer, sometimes even just a few atoms thick, of chrome onto the outer layer to make it shinier, harder, more resistant to corrosion, etc. It is added to steel to make "stainless steel". For this reason many people choose chrome in their kitchens and bathrooms. It's not good to use corrosive cleaning agents on a nickel finish.