The lightest tires have very thin casings with minimal rubber and a high thread count. These tires are efficient and supple, and accelerate almost effortlessly. But they can be vulnerable to cuts, and are more prone to rolling off the rim and feeling squirmy.
The thickest tires have two full plies of woven nylon casing material with plenty of rubber, and a lower thread count. These dual-ply downhill tires are are noticeably slower to accelerate, and make the bike heavier and less efficient, but they are more resistant to cuts, feel more stable, it's much more difficult to roll them off the rim. You can also run lower pressure on this style of tire.
Most riders are looking for the middle ground between these two extremes, which can mean a little more rubber, maybe a lower thread count or two layers of a higher thread count, and/or a protective layer other than rubber and threads (like Maxxis' Silkshield, EXO, EXO+ and other puncture technology). These tire trade offs, which also result in significant weight trade offs, are critical because the tire has to spin up to speed for the bike to get going, and it takes more effort to make that happen with a heavier tire. Also, the added gyroscopic effect that comes with a heavier tire makes the bike more resistant to rider input.
Visit the Maxxis tire tech page, which breaks down tire anatomy and explains casings and protective layers used in its mountain bike tires, and also check out Bikerumor.com's tire tech series to learn more.