These words are often used interchangeably, but they shouldn't be. Alternative is a noun meaning a choice, preferably between two possibilities, not several. Alternate is the adjective that's related to the noun alternative. Alternative should not be used as an adjective, as in "alternative choice." Here is an example of the correct use of each word:
"We have the alternative of taking the train or driving."
"The alternate method of transportation is the train."