The fresh [1] kind has a lot of water; it's typically sold either tightly plastic-wrapped box, or in a bag or box in a bath of liquid. It's noticeably squishy. If you use it as-is, your pizza will be a small sea of water. It needs to be dried a little out before use.
The dry kind has been dried (it's a little more than that, but drying is the important part) until its moisture content is around 45-52%, compared to at least 60% for fresh mozzarella, and it's either a block or shredded. The low moisture content means it lasts a long time in the fridge, and the long shefl life makes it popular in restaurants.
The difference, on pizza, is how it melts. The wet kind just doesn't melt that well; it turns into white globs with a kind of leathery exterior [2] and looks not unlike fried egg whites. The dry kind melts much more evenly [3], and doesn't have that same leathery texture.
Both types of mozzarella are definitely "real" mozzarella. The Italians even have several types of classic Italian dried mozzarellas, such as scamorza. You can also get smoked, dried mozzarella, for instance.
The true shredded abomination is the stuff that isn't cheese at all, but some horrific stuff made from vegetable oil and food dyes, sometimes in the name of being "healthier". Oh, and American cheese. There's no cheese in American cheese.
Personally, I think buffalo mozzarella is overrated. You can get good and bad buffalo-cheese pizza, and good and bad cow-cheese pizza; the big taste difference is usually elsewhere, in my experience.
[1] http://www.cheese.com/media/img/cheese/Fresh_Mozzarella.jpeg
[2] http://blogs.browardpalmbeach.com/cleanplatecharlie/grimaldi...
This is done both from a cost perspective and because the cow-milk mozzarella is not as fresh as the buffalo one (which in isolation of course tastes better). Since buffalo mozzarella is usually consumed within 24 hours of production, it liberates too much milk in the pizza.