It seems to me that men and women compete very differently.
A woman competes with other women by tearing the other women down. Women are nit-picky, looking for ways to disparage and discredit the successes of others. Women construct elaborate, ambiguous, social structures that can be used as weapons to reward their friends and punish their foes in a near-arbitrary fashion.
A man competes with other men by building himself up. Men gain stature by "going where no man has gone before": exploring new territory, building a bigger skyscraper, buying a fancier car, and so forth. Yes, it's all very phallic.
Nit-picking annoys men because to us the accomplishment speaks for itself, regardless of its imperfections. Even Christopher Columbus's enemies acknowledged the greatness of his feats. To women, no accomplishment has any value other than that granted to it by the social construct; if a woman doesn't like Columbus, she'll find a perfectly reasonable excuse to dismiss his achievment because it wasn't done "properly".
Similarly, women appreciate compliments on the details. If she changes her hair, a simple "You look great" pales in comparison to "You changed your hair, and it looks great". Men, on the other hand, want to be admired for their large accomplishments, and generally couldn't care less about the details.
Men try to impress women as if they're men, and this leads to confusion. To a man, a one-time gift of a $10,000 new car is equal to 100 weekly gifts of $100 each, and he won't understand why the woman is still expecting more gifts, frequently, (much) less than two years later. He wonders, Why doesn't she appreciate what I've done for her already? Women try to impress men with details. She buys him a shirt or makes him a birthday card, and he hardly notices because they're small things -- but she feels slighted.
Advice to men: give her lots of small gifts, frequently; express appreciation for small things; try to notice details. (I've found that noticing details is easier for me if I focus on a specific thing I know she likes. For instance, if you know she likes her fish, look at her fish every time you go over and remark about any differences. This gets you off the hook for having to notice other things that are more subtle.) If you want to compete with a woman, it's easy to tear her down by pointing out tiny flaws in her work, even though to you they are inconsequential. Yes, this is fighting like a girl.
Advice to women: recognize his large accomplishments and compliment him on them as long as they're active, not just once. For instance, he may spend every day building up a career that he's very proud of, but you may not think to express frequent admiration because there is no specific detail for you to anchor your compliment to. If you want to compete with a man, it's easy to discredit his accomplishments by comparing them in total to greater accomplishments by others; don't waste time criticising small details, because he'll dismiss your nit-picking. Yes, this allows him to retain credit for the accomplishment, but it makes him think that the credit is insignificant.