Crave means to want something with an overwhelming desire, to need something urgently. Crave may also mean to beg for something. Crave is a verb, related words are craves, craved, craving, craver. Crave comes from the Old English word crafian, meaning implore or demand.
Craven means cowardly, faint of heart in a contemptible manner. Craven may be used as an adjective or a noun, related words are cravenly and cravenness. Craven comes from the Middle English word cravant, meaning defeated. As the differing etymologies show, craven and crave are unrelated words.
Examples
The early chapters present a history of human food consumption and challenge the notion that humans evolved to crave the Paleo diet any more than any of the other diets they’ve subsisted on in the past. (The Washington Post)
This glorified grilled cheese has a shatteringly crisp exterior and the gorgeous melty center we all crave. (The Chicago Tribune)
Shocking photographs taken inside a hospital show a crazed fantasist hooked-up to life-saving machines as she lied about having cancer because she craved attention. (The Daily Mail)