Paulie is a hot-blooded Italian and Rocky's best friend. And, other than being two Italian boys, we're unsure why they're friends (the movie never explains it). But they stick together through thick and thin—mostly thin—like guys who have grown up with each other and known one another for thirty years.
A lot of people think Rocky is a bum, but Paulie is even bummier. He drinks. He yells at Adrian. He even gets violent. Even though both he and Rocky are at the bottom of their respective barrels, Rocky has the ability to crawl his way out, and he does.
Paulie doesn't. Paulie will always be Rocky's subordinate. Paulie isn't self-sufficient. He works a job that he doesn't like, lives with his sister, and rides his friend's coattails.
Paulie wants Rocky to help him out by getting him a job with Gazzo, the loan shark, but Rocky always changes the subject. Rocky never says exactly why, but it's probably because of Paulie's violent nature—he knows it would get Paulie into trouble. One excuse to crack skulls, and Paulie is likely to not only be cracking skulls, but drinking mead out of them, Viking-style.
Paulie takes a lot of his pent-up frustrations out on Adrian. He says,
"Adrian ain't sharp. Adrian is a loser. She's pushing thirty freaking years old. And if she don't watch out, she's gonna end up dying alone."
Replace "Adrian" with "Paulie" and the sentence would actually be correct.
However, Paulie isn't entirely a loser, because he's a good buddy. He helps Rocky by supporting him. He sets him up with Adrian in the first place. Rocky is poor, and Paulie gives him meat from his meatpacking job, helping Rocky keep up his strength (and get swole). And Paulie pulls some strings to allow Rocky to beat the sides of beef at the meatpacking plant in his iconic training scene.
If you're going to capitalize on the success of a friend, it helps to have a friend like Rocky. Rocky is laid back and never takes any of Paulie's drunken rants personally. Oh, and he also generously allows Paulie to advertise on his robe, earning Paulie a nice chunk of cash. Paulie's good deeds are repaid, and then some.