For the average user both versions perform the same functions (no changes in formulas, etc.); as boxtops notes the main difference is the look of the program. 2007 uses the "ribbon" to display commands rather than the menu bar, and the ribbon can't be customized as the menu bar could, making it harder to get all your favorite commands in one place. 2007 also allows vastly more rows & columns than 2003 does.
Don't overlook the existence of Excel 2010, which is essentially 2007 with customizable ribbons!