Why do Americans spell this word differently than other people do?
Because we do what chemists tell us.
In 1807 Sir Humphry Davy discovered aluminum, and named it with only one i . The International Union of Pure and Applied Chemistry soon decided to add the extra i, to parallel the endings of sodium, potassium and other element names. The two-i spelling (aluminium) was the standard throughout the world for a time, even in the USA. (This explains why Webster 1913 has the two-i spelling as standard.) But in 1925 the American Chemical Society officially reverted to aluminum, and the one-i spelling has been standard in the US since. Which is to say, aluminum is not a misspelling of aluminium, and it's not a degradation of a traditional "aluminium" spelling. Instead, the American spelling is the earlier spelling. (And, contrary to what my British friends think, Americans aren't mispronouncing the word -- we spell it differently, and that's why we say it differently.)
Update: Redalien tells me that Davy originally named the element alumium (no "in"), which conforms with the common "ium" ending pattern. Then Davy changed it to the current U.S. spelling, then the IUPAC changed it to the current spelling accepted outside the U.S. So Davy was even more odd and contrary than I thought at first!