The stats regarding less-pay-for-same-work are largely misquoted by politicians (fortunately). There is a discrepancy, but it is a comparatively smaller one and in some studies even within the margin of error. The statistic often quoted placing female wages around 70%-80% of male wages is a misstatement, as the actual stat is roughly women make 70%-80% of men's total income, not for the same job for the same hours at the same skill. That stat is usually placed in the 90% range (which while it should be 0%, is a far cry from 30%). The income gap is attributed to the disproportionate number of men in higher paying non-executive jobs (hazard pay usually), the number of men serving in executive positions, and that women often work fewer hours on average (are more likely work part time period, plus take longer leaves). I think the biggest true issue here goes to the second point in the list, in the number of men versus women serving in executive positions. The other two are largely individual, but it is a huge cultural red flag with the executive balance. I won't bother linking here, but politifacts, forbes, and countless sites have articles covering all of this from various angles and with too many different numbers to begin to get into, but in general the wage gap just isn't the end-all cultural dilemma it is made out to be and I worry largely distracts from the bigger issues regarding women in the workplace.
Anyway, that aside...
I've always been a little confused by the notion of male privilege since I never really experienced it myself. (Partly because I didn't do anything, partly because even at "most male" I was never the slightest bit alpha.) I have seen at least one major instance involving it, but then I've also seen a handful of instances involving female privilege. In the first case, it was involving a classic southern US "good ol' boys club" with divorce lawyers and a judge screwing my mom over royally. But then in general, it is often thought that women get the better deal in divorce settlements, particularly custody hearings, so I don't know if this is the best example. For an instance of the female privilege, well, recently my little sister was in a car accident that was 100% her fault. But she's a cute little white girl, and so she walked away without the slightest repercussion. She wasn't even trying to get out of it, it happened entirely on its own. This is pretty common around here apparently. (Though at the same time, is also a bit patronizing.)
For the most part I believe that there is some inherent privilege along with some inherent bad about being both male and female, and also that it may vary heavily based on race/ethnicity/religion/etc. Playing into Brandon's original post, there are countless negatives that come along with being specifically a black male in the US. Cultural preconceptions leading to higher incarceration rates, suspicion even in the most innocent circumstances, starker unemployment, and so forth, which are issues that largely do not extend to black women, at least not to the same degree. (Make no mistake, if a black male had been in my sister's place, he probably would have spent the night in county lock-up.)
An example of bias against men that has always rankled me is the way people assume that all guys are predators with children. I love kids, and kids love me. But because of how I presented myself in life as male, any interaction I had with children I had to be extremely careful about. (Often this meant playing it in such a way that made it seem as though I was inconvenienced but being a good sport about it if children spoke to me, when in reality I loved the interaction and found them adorable.) This simply isn't an issue for those presenting as a woman, which I always found very unfair.