The greatest difference you will notice is by purchasing a really nice set of rubbers to go on those rims. The 15" catagory has little to no performance tires available for cornering fun.
The 16" catagory will unlock a lot of different tire brands and speed ratings. If you have the $$ I bet you get more enjoyment out of your Mustang with a bigger rim.
Width, Sidewall and rubber compound make up the tire's ability to stick to the road. Width provides traction. Wider tires, more footprint. Thus you can corner better and also stop better. The shorter sidwalls ride stiffer and provide less feedback in the turns (I mean like 45 series or lower.) Do your homework at www.tirerack.com read as many reviews and ratings as possible.