The Godfather, Part II was first announced by Paramount when Francis Coppola was still finishing the first film. Mario Puzo was the first person to sign on for the film, but producer Al Ruddy backed out. After the Godfather was released, eventually Francis joined the bandwagon, only, however, if he had full control over the project (with no studio interference). He had his wish, and most of the original film's players returned (Marlon Brando was even due to return, but apparently he was asking too much for his small role so they simply rewrote the scene at the end of the movie).

So, basically, The Godfather, Part II was made to cash in on the original's success.

Godfather III, on the other hand, was made due to an ongoing interest by fans and many studio execs. Francis eventually agreed to return and then wrote, alongside Mario Puzo, the story which would be based on the films' original characters, as well as main headline topics.

And, if you really think about it too, "The Godfather, Part III" was also made to cash in on the first two movies' success.

So yeah it was all about money.