Mancini vs Conte
Similarities
There's no doubt that work ethic, tactical discipline and a solid defense is the basis of both manager. Its basically the ethos of every Italian manager of that generation. Mancini is far more stable then Conte but won't shy away from going hard on areas he think that are not good enough even publicly. He did so during his time at City.
Differences as a player.
First of all it comes from their upbringing. Conte joined Juventus at age 22. Once retired he returned as their manager before he hit 50 years of age. The guy was raised in environment were second best is unacceptable. Character wise Conte was always a boy scout. Lippi would say that he was the only player he never had to explained tactics to. Conte knew what needed to be done and had the tactical awareness to do it. Mancini's career was different. First of all he was a no 10. It was one of the few positions were a player was allowed to express himself in a creative way. Secondly he always played in an underdog team whether its Sampdoria, Lazio or Bologna. Mancini was also a bit of a rebel. His prank on Sacchi costed him the WC of 92. If I had to compare the two it would be like comparing Keane to Bruno.
Difference as a manager.
Conte's career is divided in two. In the early part he managed small fry clubs. The stakes were low there and he stood out of the limelight. Then he moved to the top league and never looked back. That's were his uncompromising nature and his reluctance to accept second best could shine and in sometimes lead to self destruction.
Mancini's career was different. He started with Fiorentina and then moved to Lazio. These are ambitious clubs with high expectations, tough fan base and small budgets to work with. Fiorentina in particular was a nightmare. They were in deep financial crisis, they were selling their top players left right and center and sometimes they didn't had money to pay the manager.
Thus made Mancini more adaptable and compromising then Conte, which surprisingly played against him. Mancini was in fact one of the few managers in football to get sacked despite winning 3 Serie A league titles on a row and would go on for most of his career having to look at his shoulder (whether its Mourinho's or Pellegrini's shadow)
Conclusion
Conte is a far bigger name then Mancini and tactically he's better. However I find Mancini to be more suited for us then Conte is. First of all he's adaptable. He can play various formations (4-4-2, 4-3-3, 4-2-3-1 etc). Secondly his default formation (4-3-3) suits our game better then Conte's 3-5-2 system. That means that players like Sancho, Rashy and co can still have a great career with us in their natural role. Mancini is more board friendly then Conte but he's no yes man. He criticized City in public and he left Galatasaray when they didn't kept their word regarding the club's transfer policy. On top of that he's not afraid of giving a chance to young players. Under him the likes of Zaniolo and Tonali were given their national debut before making their debut in the Serie A.
Mancini can't care less about money. However if we sell him a great project then he'll probably take the job.