No. The 256 MB on the $279.99 Xbox 360 will play Halo 3 the same as the 20 GB Xbox 360. But you save A LOT of money if you buy the $349.99 Xbox 360 because it comes with plenty of accessories you would have to buy for retail price if you just buy the $279.99 Xbox 360. But if you're strapped for cash, get the $279.99 one.
But if you're going to play Halo 3, then you're going to play on Xbox live because it's just awesome. The $349 Xbox bundle comes with a FREE headset and FREE Ethernet cord. It also comes with two games which you can TRADE-IN after you buy it at your local Gamestop.