Regarding Depleted Uranium:
It's got a pretty high density (18.06 grams per cubic centimeter - compare with lead at 11.34 g/cc) which makes it good for bullets and armor and the like.
Even better, it's CHEAP. Lots and lots of effort was put into making weapons-grade uranium during the cold war. To make weapons grade material, you take a whole lot of uranium, then go to lots of trouble to seperate out the U235, which is less than one percent (.718%) of naturally occuring uranium. U235 is the stuff that makes the big boom. What's left is mostly U238, and useless for bombs. So after making a huge pile of atomic weapons, something had to be done with the even huger pile of "worthless" uranium. The result was DU ammunition - the DU is both radioactive and chemically toxic, which is why you can't go bear hunting with it - just tank hunting.
Addendum:
As I understand it (according to Tom Clancy's Tank book), tungsten is also used in anti-tank munitions. Tungsten has a slightly higher density than depleted uranium, so theoretically should be better for that purpose. So I guess it's a cost issue. When a uranium sabot hits a tank, the uranium powders and ignites from the friction of penetrating armor. Uranium should react a lot like aluminum - think thermite - so the inside of the tank gets showered with white-hot metal sparks. Not too comfortable for the crew.
U238 isn't completely useless for bombs - you can make plutonium out of it fairly easily. If I remember correctly, the first atomic bomb test, and one of the bombs dropped on Japan were plutonium-based.