Robbie is a socially awkward boy who doesn't seem to do many things right. Rex is a smooth talking, Smart-Alec, cool guy who is really a puppet. And that's all he is. So how is Rex such a diverse character from his puppetmaster? Here's how I see it, due to my constant way to overanalyze and overthink the simplest things. Once I made this, I believed it would be something I would love to share.
Robbie knows he can not do much right, such as talk to girls or act cool. It should not mean much to people, but in Hollywood Arts, it apparently means a lot to the poor guy. Yet he can not really do anything about it, probably since he's been like so his whole life. So he copes with it in the strangest way. he is trying to help himself through Rex. How he discovered this method, I have no idea. But it seems effifient enough for Robbie to be useful everywhere.
Rex is the thing that makes him feel more confident, though it is obviously hard for one to tell. And even it means bagging on himself [Robbie], for he believes it will make him a better person through Rex's harsh words. And the way Rex talks is actually Robbie, and he couldn't say this because either he wants to be nice to his friends to say something rude or painfully truthful or he's too socially awkward to speak up. So though it may seem weird to always have a puppet around you all the time, it secretly helps him through his daily life. Someday, he will have to grow up and give away the puppet to be his own person. He has been so close to Rex, however, so I haven't a clue when that day will come.
Summary: Rex is Robbie, and Robbie is Rex. They're the same character, despite their personality clashes.