I belong to a few, although I'm not sure there are equivalents to these in Britain.
1) Phi Kappa Phi is an honour society for students who graduate in the top few percent. I was offered this one after completing my MBA. It looks good on a CV, and I did get an article published in their magazine. I'm able to reach out, with some credibility, to other members.
2) Beta Gamma Sigma is another honour society, this time for top graduates from business school. The local chapter organises monthly meetings led by a top business leader. They've had some really good people talk, and it was a privilege to meet them, plus I've made some good contacts by networking. Again, this one looks good on a CV.
3) Technology Association of Atlanta. For networking, this one is extraordinarily good. They have about 20 sub-topics - HR, technology, venture capital, etc. - so I could attend a meeting almost every day if I wanted to. They organised the startup competition we entered last year, and they hold several events designed at pulling together entrepreneurs and financiers. They also hold professional training sessions on a variety of topics. The opportunities for networking are really excellent, and I've made some very relevant contacts.
People have mentioned annual membership fees of the order of £100 or £300. That's very high. The two honour societies are about £25 (although I'm a lifetime member of one of them). Even TAG is only £150, and they offer a tremendous amount in return.