Progressive get progressivly stiffer as they compress. So it may start as a 200lb/in spring but firm to 250lb/in spring when compressed. Specific is only one rate, which will result in a harsh ride and have no performance increases.
Specific rate springs are a constant rate. Progressive are softer at first, then as they compress more (like in cornering or a hard launch) they get stiffer.
Specific is stock, with the same distance between coils, and is always compressed to some degree close to it's bottoming out by vehicle weight. Progressive is one or more coils are closer together, making for a softer ride in real world driving, but still firm in the rest of the spring for hard driving.
Specific are stiffer, but prone to bottoming. Most road racers use progressive because, even though they are floaty, they follow the road better than a stiffer spring. But these guys are racers and know how to use the progressive spring to their advantage, as they are much more difficult to handle than a specific rate spring (diving into turns, as an example).