Islamic is the more general adjective, meaning "of or pertaining to Islam". This would apply not only to Islam in the modern world, but the historical Islam.
"buildings in the early Islamic style", "Islamic history", "The Government is seeking the active participation of the Islamic community in the formation of its race-relations policy".
Islamist has a more restrictive meaning. It is only used in the modern context, and is used of persons or organisations who actively seek the resurgence of Islam through political or military means. It is principally an adjective ("an Islamist organisation", "leading Islamist thinkers", "XXXX is an example of an Islamist group with both a political and a military arm".), or as a noun ("many Islamists wish to see sharia law predominate").
That is my take, anyway. Being something of a touchy subject for some, there may be those who disagree with my definition of "islamist", but I think most would accept "Islamic" as I have used it.