In addition to the family ties between Paul and Carlo,my understanding is that Paul had been basically an acting Boss in the last few years of Gambino's reign.He was mentored in the finer points of Mafia business by the best,and I believe that included how to ease the pain of any perceived snub by Neil.

My personal opinion is that Neil really didn't want the top spot,because,in effect,he already had his own Family.Even if he wasn't in jail,I think he would have supported Paul,and told his underlings that it was the way of Cosa Nostra. This way Neil wouldn't lose face,but on the contrary,would have been held in higher esteem because of his "loyalty" to the Family.

As a sideline to the whole blood ties thing,not only was Carlo Paul's brother-in-law,but Gambino's mother and father were first cousins.