Discomfit means to defeat, overthrow, thwart, frustrate the plans of, and to make feel uneasy or embarrassed. In its first sense it is an absolute word—it should not bear qualifying adverbs. Thus prefer His plans were discomfited to His plans were wholly discomfited. Its latter sense likely arose through confusion with discomfort, though that sense has been standard throughout modern English and should not be considered incorrect. Discomfit becomes discomfited and discomfiting in both American and British English.
Discomfort, a noun, refers to a state of unease. Though its verb form exists, it originally meant to deprive of courage and wasn’t used as a synonym for discomfit until well after the latter was an established standard. As such, reserve discomfort as a noun and discomfit as its verb equivalent.