In my opinion, the difference between "Félicitations!" and "Bonne continuation!" is that the former is used to congratulate someone and to celebrate an achievment such as passing an exam for example. The latter; by contrast is usually used to express heartfelt encouragement to someone who is undertaking a project and who has reached a milestone towards its accomplishment.
I am having a hard time trying to find a proper way to find the English equivalency of "Bonne continuation!" so, any help would be highly appreciated.