track tires vs roadtires

a big difference is going to be durability.

road tires have to deal with much more uncertainty so usually have more plys or puncture proofing layers. this will usually make them heavier and less "supple"

As you probably know, track-specific tires are usually race tires. Since the track racing surfaces are much smoother and better maintained than most roads, track tires don't need the durabiliy of puncture-resistant casings and thick, long-wearing treads that road tires have.

Track tires are designed to be as light as possible (140g for a 700c tubular track tire is not uncommon) and handle very high pressures (200+psi) to minimize rolling resistance. The high pressures usually negate any gains in suppleness from the thinner tread and casing.

A well maintained concrete track like Trexlertown probably won't mean a big difference in tire choice compaired to a wooden velodrome. Maybe a slightly thicker tread would be needed to deal with the slightly rougher surface.

For the vast majority of people, a good set of road clinchers, like Continental's GP3000 or Vredestein's Forteza Road, pumped to max psi are fine for track racing. If you want tubular tires, Continental's Sprinter or Challenge's Criterium are good places to start.