My take, I don't think there was competition, suspicion if any
What did Don Vito Corleone see in Rocco LamponeExtract: So, just as Neri is Michael's Luca, Rocco is Michael's Clemenza
The only scenes, from memory -
The hooker murder / Geary compromise
The new Don Tom asserting his authority! Dead Tahoe assassins
Tom: Alright, get rid of the bodies
Rocco: Where's Mike?
Tom: sternly, Rocco! [Then Tom walks off]
Rocco wasn't happy! taking orders from Tom
The Desert Inn - when Michael said to Neri and Rocco to step outside, what I found really interesting was neither Neri nor Rocco engaged in any conversation whilst waiting to get back in. It was like they both avoided and kept their distance from each other!