I'll nutshell this
Basically, nickel is warmer with a more midrange and stainless steel is brighter with scooped midrange.
Now, not all nickels and steels are created equal.
For instance, DR Sunbeams and D'Addario XL's are both nickel, but they sound different. Sunbeams are warm and supple with a nice 'snap' and XL's are brighter and sound 'clangy'. DR's also have a smooth feel because of they way they are wound. XL's are not as smooth, and Nickels like Rotosound 66 and Blue Steels are quite rough..but still smoother than stainless steel.
Stainless steel strings have a rougher feel, and some say they chew up your fingers and frets. They have a more metallic, in your face bright sound. Again, there are differences. Take DR HiBeams and Rotosounds. DR's feel smoother and have a more even tone, while Rotosounds are the roughest strings on the planet and have a very trebly sound.