Nickel plated vs pure nickel guitar strings
Yes, there is a difference between the two. More so than you might think.
Let's first define what the difference is between these two string types.
Nickel plated guitar strings: Plain steel strings for the high-E, B and G, and wound strings for the D, A and low-E where the core is steel and wrap around the core is nickel plated.
Pure nickel guitar strings: Plain steel strings for the high-E, B and G, and wound strings for the D, A and low-E where the core is steel and wrap around the core is all nickel.
"Pure nickel", as far as I'm aware, never means the strings are absolutely 100% pure nickel. Rather, it is only applied to the wrap around the wound strings. That wrap is what gives wound strings labeled as "pure nickel" a different sound.
Yes, this means the pure nickel tone only concentrates on the D, A and low-E. The plain strings are still the same steel as they are in nickel plated sets.
Side note before continuing: If you want strings where the steel is truly different for the whole set (apart from the coated offerings many string companies offer), Ernie Ball makes two you can try, Stainless Steel and Cobalt.
What makes a pure nickel wrap different on the wound strings?
Answer: Less treble response from the wound strings.
This sounds like a bad thing, but for some it's exactly the type of sound they want.
Wound strings always ring louder than the plain strings because there's more metal moving around. It is typical for many guitar players to lower the bass side of the pickup and raise the treble side so the thinner strings are heard more clearly. But sometimes, no matter how much adjustment is made, those wound strings just ring too loudly.
This is an example of where pure nickel wrap can save the day. The nickel is softer and the decreased treble of the wound strings allows the plain strings to be heard more easily. The end result of this are chords that sound "fuller".
Good for everything but metal
Pure nickel wrap wound strings work for everything but metal. Metal guys want that "sounds like piano wire" level of brightness (such as from D'Addario ProSteels) that pure nickel wrap strings absolutely do not have.
If you play metal, pure nickel wrap strings are not for you.
But for everything else, the pure nickel wrap wound strings work quite nicely.
How to know if you want the pure nickel wrap wound strings or not depends on the answer to the following question:
Do your D, A and low-E strings ring too loud no matter what adjustments you make?
If you answered "yes" or "sort of", then give the pure nickel wrap strings a try.
Know that there is not a gigantic night-and-day difference in brightness between nickel plated and a pure nickel set. But it might be "just enough" to get the wound strings at a volume level that better agrees to your ears.
On a final note, yes, D'Addario also makes pure nickel sets if you'd rather use that brand.