We all knew that Micheal went to see Cardinal Lamberto because he wanted to reveal the corruption, however, the confession was not told according the film to be part of the interview.
Then the question arises. Why did Micheal make the confession, out of repenting? This may be reasonable, but Lamberto would then know who Micheal was. How would Lamberto agree on Micheal's control over the corporation to purify his money? Micheal did something dangerous?

Lamberto already knew who Michael was. And Lamberto was a true man of GOD and therefore would never reveal Michael's personal confession. As a man of the cloth he is bound to silence when hearing a personal confession. He saw this attempt to confess by Michael as a sincere attempt for redemption.

Lamberto would also have attempted to clean up the whole Vatican scandal, and that is why he was murdered. As Michael said, he was a good and honest man.

Michael wanted this whole Immobiliare Bank / Vatican deal to be a legitimate deal. It was another way for Michael to redeem himself from all the illegal business dealings that he had done in the past, and to fool himself into believing that he had brought the Corleone name a step closer to legitimacy.

It's what GFIII is all about; Michael's trying to finalize his attempt at legitimacy. But no matter how many legitimate business deals he may have done, the illegitimacy was really something deep within himself, in his own subconscience.

"Just when I thought I was out, they pull me back in."

An excuse. He was in to deep already because of the horrible things that he did in his past, especially ordering the murder of his own brother. So unless HE himself could find redemption for his sins, could forgive himself, he could never really get out. And when it looked like he finally did, he paid a steep penance for his sinful past ; He lost his daughter. His dealings and sinful acts came back to haunt him and indirectly caused his daughter's death. And Michael was sentenced to live with that guilt until the day he died.

WTWT, you've raised an interesting question. The nominal reason for Michael's confession was for FFC to provide us with a dramatic and emotional scene--for once, showing Michael crying and seeming repentent. He may actually have been repentent in that moment, though, as Vito said in the novel, "A man has but one destiny," and Michael chose it.

I agree with just about everything Don Cardi posted in his excellent analysis. But it raises another question: what did Michael really intend for Immobiliare? DC succinctly expressed the contradictions in Michael's life when he posted:

"Michael wanted this whole Immobiliare Bank / Vatican deal to be a legitimate deal. It was another way for Michael to redeem himself from all the illegal business dealings that he had done in the past, and to fool himself into believing that he had brought the Corleone name a step closer to legitimacy."

"Fool himself" is right. Michael constantly strove for "legitimacy" (and not just in GFIII), but on his terms, not by others' definitions. I don't know what he intended for Immobiliare. It's one of the major weaknesses of GFIII's plot. I have never been satisfied with any explanations of why he was so fixated on Immobiliare (as opposed to some other investment or business enterprise), why Lucchese opposed him, why Archbishop Gilday conspired with Lucchese--to say nothing of who Altobello was, and why he conspired with Zasa and Lucchese to get rid of Michael.