Expedient describes something that is suitable, an action that is appropriate for the situation. Expedient may also have a negative connotation, it is sometimes used to describe an action that is advantageous or convenient for the situation, but not necessarily the most just or moral action to take. Expedient is an adjective, the adverb form is expediently and the noun form is expediency or expedience.
Expeditious describes an action that is done quickly and efficiently, the word expeditious does not have any negative connotation. Expeditious is an adjective, the adverb form is expeditiously, the noun form is expeditiousness. Remember, expedient may be considered to have a neutral or negative connotation, expeditious has a neutral or positive connotation.
Examples
One parent mentioned teacher evaluations, and Fagen responded by saying that she is reviewing the new teacher evaluation system that Texas came forward with; she said that in her experience, evaluation systems like this are a lot of work, so she will be looking at the best and most expedient way to implement it. (The Tribune)
From the pixelated accretion of contradictory assertion, an image slowly emerges: of a continent repeatedly bent to the will of colonial prejudices, until its leaders eventually found it expedient to internalise those prejudices into a narrative of defiance against the decadent imperialist culture. (The Telegraph)
His wife, perhaps because it was politically expedient, was his chief defender throughout. (The Pocono Record)
Terming flogging of Dalit youths in Gujarat as “a stigma on mankind,” Union Home Minister Rajnath Singh on Thursday assured expeditious inquiry into the incident and said the probe has been handed over to the CID crime branch of the state police. (The Business Standard)
Against the backdrop of Vijay Mallya episode, the Lok Sabha passed a bill on Monday that provides for expeditious recovery of bad loans by the banks, with the government saying the country cannot have a banking system where people take loans and do not repay. (The Hindustan Times)