"Implement" does not indicate completion. It means "put into effect" only. For example, you could say that Microsoft has implemented a language translator in their word processing program. It just means the translator is there now. Maybe it works, and maybe it doesn't. While they are working on that project, you could say they "are implementing a translator". If you want to say that they have finished adding the translator, you need to say "they have completed the implementation of the translator".

Hello Irina, The difference is that the former means “осуществленный проект” and the latter “законченный/завершенный проект”. As you can see, we also use different words to describe those expressions in Russian. In general, to implement something is to carry it out and complete something is to finish it.