I've been grading some Moot Court briefs, and this one came up.
Incite comes from the Latin "in" for "in" or more realistically, "to cause a person to be in" and "ciere" for "to set in motion" through a Late Latin derivation "citare" for to summon to a church court (related to citation). So it is not a big leap to get to the current usage of "to cause a person to be set in motion" or "to stir, encourage, or urge on; stimulate or prompt to action". Meanwhile, inspire comes from the same Latin "in" and "spirare" for "to breath", so quite literally, "to breathe into". It was originally meant as to breath life into, and then "to give rise to" like breathing life not just into a physical body, but into activities, and then their ideas, and then all intangibles. The initial usage has been abandoned mostly, but all the others remain in varying degrees. So pretty much, you can inspire anything.
So, what is the difference. Motion v. breath (life). Hmm. Well, incite can only be used with an activity as from the etymology, whereas, inspire is broader. English teachers incite reading with summer reading lists. English teachers inspire reading? Maybe, but not really. English teachers inspire writing novels. Incite requires an impetus--a deadline or an adjudication or guilt to motivate the action. Waiting for the opposition to my motion, my call to Plaintiff's counsel finally incited him to send it to me. Having promised his mother that he would clean his room, the threat of being grounded incited him to actually do the work. Inspire requires a new thing come from the action. While writing my opposition to Plaintiff's counsel's motion, I was inspired not just to attack it on the substance, but also on issues of bad faith. While cleaning his room, he was inspired to wash the car and take out the trash as well. Yeah, like that would ever happen. As for things beyond activities, the rousing cheer of the fans inspired the rookie with confidence to hit yet another double. In am not infrequently uninspired with any ideas for sentences using the wotd. I may be inspired to incite Plaintiff's counsel to be a better lawyer. However, I cannot incite Plaintiff's counsel to inspire his client to settle.