I don't mean this in a rude way, I'm honestly a bit confused - have you watched all Mad Men episodes? I just don't understand how anyone couldn't know why Pete is the way he is, you may lack sympathy for him, fair enough, but how can you not know why he's so insecure in himself? Whole episodes have centered around his issues!
He's insecure for so many reasons, which other posters have already spoken of. His family don't think much of him, he's "Peter" to his parents, where-as his brother is "Bud", that alone shows how Bud is favoured and more affectionately thought of by the parents than Pete is. His parents were both emotionally absent, harsh and pompous, and he's never been able to be the son they want, he may have been born with a silver spoon, but he's socially awkward, he can't be naturally charming or good with people - there's something about him that people inherently take against, whether he's deserving or not. He's got a more modern, free thinking out look, in some areas, than his family and other WASPs have, which also marks him out. Basically, he's a black sheep, of his family, within the office - it's the thing that first drew him to Peggy and Peggy to him, they both recognised within each other that quality; that part inside that sets them apart from the rest of their social circles, they just don't quite fit, and have to make an effort
to fit in (evidently Peggy is much better than Pete at this).
He has a chip on his shoulder for many reasons. Because he can't be inherently charming and win people over like Don can, and he knows he'll never be able to. Don may have had a worse start in life, but now, in many ways, he glides through it, charming his way out of anything - where-as Pete has it harder, he has to work to get people on his side, struggle to adapt to people socially, as well as constantly being put down and emasculated by people at home and at work.
I think looking at Don and Pete together most clearly illustrates how two people can come from very different backgrounRAB, and how - if you were to look at them on paper, you might think 'Well, evidently this Pete guy is going to be better off and happier', but he isn't. He's just as messed up inside. In many ways, he has it harder than Don does now, and although his upbringing was cushier, from what his parents seem like, it's easy to imagine his childhood was lonely, emotionally deprived and heavily repressed.