The newer remake has, as you might expect, a much bigger budget, more explosions, and a much better perhaps-requisite fight in the garage. In star power, it's probably a tie as the new one has Jason Statham and Ian McShane (Lovejoy? Anyone?) but the original had David Carradine, Sylvester Stallone, and The Real Don Steele. But the biggest difference, I think, is scope and breadth of vision.
The remake is set just a few years in the future (2012) and the entire race takes place in a near-future privatized prison. (I'm intrigued by the notion that privatizing prisons, something that already occurs, would allow those corporations to just kill off prisoners at whim, but ok, sure.) Jason Statham is an unalloyed good guy -- framed for the killing of his wife and wanting to escape to be re-united with his daughter.
The original was much more ambitious. It's set a full 25 years in the future, when the United States, which now extends to Asia, has been ruled by a fascist "Mr. President" for most of that time. The eponymous race is cross-country and the racers are deviled by the attacks of a resistance who are targeting the race as a symbol of the government. For all of the cheesiness and over-acting, I found it an actually engaging story.
Death Race: B-
Death Race 2000: A-