Power is the ability to generate as much force as possible in a split second.
Strength is the ability to exert a maximal amount of force regardless of time.
The difference between someone who is powerful and someone who is strong is marked - here is a practical explanation:
Let's say person "A" = powerful, and person "B" = strong.
If you put both guys in a gym and got them to deadlift (
http://tinyurl.com/qjfmtxj) as heavy as they possibly could person B would be able to lift more.
It doesn't matter how quickly you lift the weight in a deadlift strength is more important.
Now, if those same guys then go outside and you give them a 10kg shot-put, person A will throw it further than person B.
Even though person B can generate more total force, throwing an object requires him to generate that force in a very short period of time.
So in the context of rugby, power translates to things like ball carrying and tackling, which happen very quickly.
Whereas strength translates to scrummaging and mauling.