New vs used?
Short Term Changes:
Mold releases compound sometimes causes traction issues, but that last at most a few hundred miles.
The first few times a tire is heated up, the curing process may continue until it is stable. This is more an issue in racing than with street usage, but sometimes this may come out as fuel economy. This would probably last at most a thousand miles.
There are sometimes excessive amounts of antioxidants and oils, and they will migrate to the surface. This causes a discoloration, particularly with white sidewall tires. This may not appear until a thousand miles and will continue for another 6 months to a year. Cleaning the discoloration off is only a temporary fix until the tire stabilizes.
Long Term Changes:
Tires obviously wear. They may not wear evenly - and this sometimes results in noise. I am of the opinion that you can trace ALL tire realted noise generation that wasn't there in the beginning to alignment - keeping in mind that "In Spec" means "some distance away from optimum"
But if the tires wear evenly, then the tire generally gets quieter as time goies on. The less depth of the tread shortens the height of the thing vibrating, so it doesn't vibrate as much.
Less tread = better fuel economy. All other things being equal. new tires will consume more fuel than worn out tires.
Hydroplaning resistance goes down as the tread wears.
Dry traction? The problem here is that as rubber ages, it get less flexible. It is possible to feel this change as both dry and wet traction (non-hydroplaning), but it is also possible, you won't feel this at all.
Overall ride quality will deteriorate. Rubber has lots of built in damping and worn out tires have less rubber = poorer ride quality.