Okay so in 1986, Optimus Prime perishes in the animated movie and his life essence/spark goes into the Matrix of Leadership which he passes to Ultra Magnus. Magnus then loses the Matrix to Galvatron, Hot Rod then leads the charge against Unicron, faces Galvatron in one on one combat and grabs the Matrix to become....
.....RODIMUS PRIME! Anointed as the new PRIME Autobot and leader by the Matrix speaking with the voice of OPTIMUS PRIME himself.
Ergo he has shown his worthiness as the chosen one and was judge as such by Optimus Prime.
So Rodimus easily handles Galvatron and then blows up Unicron, then leads the Autobots to retake Cybertron amidst the confusion and chaos of the fight against Unicron.
Now at first he seems confident and in command. All the Autobots accept him as leader since the Matrix has chosen him.
like they are a bit too devoted to a power object whose origins are basically UNKNOWN.....
Then in Five Faces of Darkness, Rodimus was still confident for the most part though he did have a moment to ask himself "Why did I have to be the chosen one?". he still led them well and later realized the answers about the Quintessons are in the Matrix and risks himself to travel in it for a chat with the ancients including OPTIMUS PRIME
After that adventure and the Quintesson plans defeated, Rodimus was then shown in some tales as a whiner and others as confident and in command.
The Ultimate Weapon is an example of Rodimus having a crisis of confidence
Fight or Flee when he led a charge to get the Decepticons off of planet Paradron. after seeing that it would be a bad war of attrition for all involved he then orders Ultra Magnus to blow the planet up. Now that is an order I can't see Optimus Prime giving which shows there is a difference between them.
Rodimus Prime was still a bit reckless and gung-ho at times just like when he was Hot Rod but that was still tempered by the power of the Matrix and the wisdom it contained.
Then we get Dark Awakening when Optimus was revived by the Quintessons as undead Optimus as part of plan to get the Matrix away from Rodimus Prime and destroy all the Autobots.
Well that is when we see that Rodimus is still reckless and eager to give up being leader. He happily shoves the Matrix back into Optimus and reverts back to Hot Rod. This was the first time this happened and thus Rodimus Prime has a major weakness in that if the Matrix isn't in him or NEAR him he will revert to Hot Rod. Kind of odd as one would think the Matrix would have permanately altered him like Unicron did with Megatron.
Anyway Hot Rod sucks it up, faces Optimus and in that fight he was Rodimus again and gets the Matrix returned once Optimus was free of control. As further proof that Optimus was meant to STAY DEAD, he sacrifices himself to save them all and is disintegrated in a super nova.
then come more episodes that alternate between Rodimus being confident or a whiner.
Then comes Burden Hardest to Bear: he is again breaking under the strain of leadership. Loses the Matrix in accident and reverts to Hot Rod who refuses to get it back. meanwhile SCOURGE of the Decepticons gets powered up and crazed by the Matrix and takes over the Decepticons and leads an assault on earth.
Hot Rod again sucks it up, defeats Scourge and reclaims the Matrix.
One can easily surmise that the Matrix deliberately did what it did to get back to Hot Rod as well as help to teach him his responsibility.
Then comes a retconned Dark awakening with a narration at the end announcing Optimus is returning.
Then comes Return of Optimus where despite the events of Burden Hardest to Bear, Rodimus is again crying about how he isn't good enough to replace Optimus. Then the spore plague is released, he is infected and later we get a Rodimus vs Optimus fight again, only this time Rodimus clearly loses and gets decked with ONE PUNCH and optimus reclaims the Matrix.
The Matrix is drained and thus no more Rodimus Prime.....
Honestly now, Optimus was basically the Captain America of the Autobots, the perfect leader always calm and in control. We never saw episodes of when he first took command and had to accept being leader, and you just KNOW that he wasn't thrilled at first to be the bearer of the Matrix. Optimus grew into being a leader over time.
That was time that RODIMUS PRIME never received. his behavior to me was perfectly normal. Of course he'd have some doubts and would want to cast off the mantle of Leadership but he also knew that no one else really wanted it or was chosen by the Matrix like he was.
But there is no doubting the courage of Hot Rod/Rodimus. It was HOT ROD that faced Galvatron one on one inside the belly of UNICRON and emerged as RODIMUS PRIME.
Rodimus had plenty of courage but he was new to being leader, and despite the fact that the wisdom of OPTIMUS PRIME was within the Matrix and hence within him, he was still plagued by doubts, which is NORMAL.
Show me a new leader that has doubts about their ability at first and I will show you a leader that doesn't exist.
Rodimus was a bit more prone to action and ordered a planet blown up to stop the Decepticons. He also made a deal with Starscream's Ghost to attempt to destroy Galvatron. I can't see Optimus ordering a planet blown up, though to stop Galvatron he too may have agreed to work with Starscream given the threat level Galvatron posed.
Rodimus was really the only one with the might to face Galvatron, especially after Galvatron was power boosted after the movie by those plasma fields he was trapped in, as well as driven insane. (now how Galvatron couldn't free himself......)
The only other Autobot I can see as being strong enough to face Galvatron would have been Metroplex.
So yes their personalities were different as they should have been. i have no doubt that in time RODIMUS would have overcome his problems and grown into an excellent leader.
Yet the fans all cried "wahhhh! Mommy! I want OPTIMUS PRIME Back!" and Hasbro turned chicken and brought him back.
Also the return of Optimus Prime caused them to do a quick edit on the GI JOE movie so that DUKE would live despite being speared in the HEART by a deadly snake!