Something expedient is helpful to you. If you vote your friend in for student body president just because you know she'll hook you up — that's an expedient choice. But expeditious is speedy, like your expeditious exit from the school cafeteria after you spilled spaghetti and meatballs all over your shirt.
Expedient describes a politically advantageous choice. Expedient also describes something that's good for you or something that's useful. It dates back to the 1400s and has its roots in Old French (expedient) and Latin (expedientem). Here are some expedient examples:
Should the government, which promised to protect airline passengers last year, do the expedient thing — or the right thing? (Seattle Times)
But Mr. Kim's defiance, at times unshakeable, has been known to soften when politically expedient. (New York Times)
On the other hand, expeditious isn't up to anything, it just means speedy or prompt. Expeditious entered English in the late 1400s via expedition, which also has roots in Old French and Latin. If you want to speed things up, use expeditious:
Mr. Fine remains hopeful of a credible and expeditious review of the relevant issues by law enforcement authorities. (Seattle Times)
The bill was given an expeditious passage as the Senate suspended order 79 of its standing rules to amend the National Minimum Wage Act. (All Africa)
Although expedient and expeditious come from the same Latin root word for "to make ready or to prompt," they parted ways by the 1600s, when expedient became self-serving. Use expedient for "advantageous" and expeditious for "speedy," like how fast you plan for an expedition to Antarctica, or across the street.