Phi Theta Kappa
Most important benefit of any kind of student org like that is networking. Nepotism with a decent resume gets jobs a lot better than a good resume.
And this doesn't just apply to honor societies. Social/service fraternities/sororities are actually probably better, especially if they're national/international. When you can have an automatic connection to someone who has good connections themself, that can be very beneficial for finding a job. Basically, anything you can do to avoid submitting a resume/application online that's going to get looked at for 10-20 seconds before someone has to make a decision on it.
The last thing you talked about are fraternities and sororities. Those are basically just student groups with a faculty wrangler. They have a very tenuous relationship to honor societies in that they sprang from the same roots, but Greek LifeTM on campus is basically just a way to get to know people and always have folks to hang out with -- mostly drunks who sleep around and spend their time partying instead of working towards graduating. They're the sort of people who spend seven years pursuing a business management degree on their parents' dime while spending all their money on liquor and skipping class.
Some, but not all. Varies on the fraternity/sorority and the university. They will also probably have a hidden body of notes, past exams, past homework solutions, an extensive alumni network with people who tend to be social, outgoing, and likeable - you know, the kind of person you want giving your recommendation, even if it's also the kind of person you absolutely despise. There are, in fact, benefits other than booze and party access, though there's also plenty of that.