The difference between a race tire and a street tire is primarily in construction.
Street tires are made with much softer sidewalls and more flexibility for 2 reasons:
1 - to allow more riding comfort (feel less bumps)
2 - to allow the tire to heat up quicker
Race tires are much stiffer by design, to hold up to the extreme forces applied in race cornering/acceleration/braking. This also is done to regulate the heat in the tire.
A race tire will work on the street, but provide a harsher ride.
A street tire will work on the track, but will not last nearly as long.
The benefit you will get out of going to a race compound is more in longevity of the tire life than of pure traction, although the two coincide very much.
If you frequently do track days, I'd recommend picking up a pair of DOT race tires. The only caution you need is that they may take longer to heat up on the road, and in certain instances have a tread pattern that doesn' t channel water as well as a street tire.