Forceful means powerful, assertive, effective. Forceful is an adjective which may describe a person or argument, among other things. A derivative is the noun, forcefulness. Forceful comes into use in the 1570s.
Forcible means accomplished through force. Forcible is an adjective which means vigorous, strong, effective. Derived forms are forcibleness, forcibility, forcibly. Forcible came into use in the early 1500s from the Old French forcier, meaning conquer by violence. In general, forceful describes a person or thing that is powerful, forcible describes the use of power.
Examples
Mr. Obama made a forceful defense of diplomacy but also castigated Russia by name multiple times in his speech for its defense of the Syrian government, its takeover of Crimea and its actions supporting Ukrainian rebels. (The New York Times)
If Warren Gatland and Stuart Lancaster were asked if they wanted to swap places with the All Blacks, the answer would be a forceful yes without a hint of hesitation. (The Otago Daily Times)
His rugby was again direct and forceful with a couple of magical touches. (The New Zealand Herald)
It took courage and determination to implement such a forceful policy given the protests from human rights groups and our regional neighbours. (The West Australian)
Can’t we think of an equally forceful qualifier for Bangladesh which will truly define what we are? (The Daily Star)
Syracuse University might kiss its kiss cam goodbye after a letter to local newspaper complained that the camera encouraged forcible kissing. (TIME)
University at Albany police are urging students to be alert after a forcible touching incident reported by a student at The College of Saint Rose campus. (The Albany Times Union)
Taipei defends forcible demolition of man’s home (The Taipei Times)