Just to insert a few more facts:
First, uranium is relatively soft -- however, that doesn't matter when it's traveling at several Mach numbers.
Second, while it is one of the densest materials going, it's not the densest. That honor goes, depending on who you ask, to either iridium, or its neighbor on the periodic table, osmium. These are not used in tank shells for the following reasons: they are rare and expensive; they are extremely painful to machine; and they have very high melting points, making them difficult to cast.
Uranium is difficult to machine because it is pyrophoric, but it is within reason.
The critics of DU do have some points -- it is slightly radioactive and it does have heavy metal toxicity. As a result of public pressure and cost issues, the US Army has switched back to tungsten anti-tank projectiles, which are almost as effective and much cheaper. They use incendiary material at the base of the shell to replace depleted uranium's incendiary effects.