The company in effect pays the tax on the dividend, not the individual but fundamentally, if you look at the whole thing together the tax liability of taking cash as dividends rather than remuneration is completely unchanged.
Although there is a potential cash flow issue in that any additional tax required on dividends (if you are a higher rate taxpayer) doesn't become payable until many months later depending on the timing of the payment (a dividend paid on 6th April would mean you wouldn't pay the additional tax for almost 22 months) . Had it been paid as remuneration that tax would have been payable immediately.
The main reason people go the dividend route over salary is, as stated above, to avoid the National Insurance which in effect makes a further 20% tax or thereabouts. You do need to watch for IR35 implications as stated though.
Oh, and missspeedy23, you can pay a dividend as often as you like. I have clients who can and do pay monthly dividends. And the next limited company I deal with who actually "hold a meeting to discuss dividends" will be the first one to do so. Most of them don't bother to issue dividend vouchers either though you are right that technically they should. It's hardly a hardship though, Ten minutes on a pc would draw up the first one and 30 seconds a time thereafter would adapt it for every one you pay.
Buildersmate, I don't see why accounting to HMRC for PAYE would necessarily be relevant to be honest? If they have other staff already they will already be doing that and if they don't and are only paying themselves through dividend and not payroll then all it will be is a bunch of nil returns. What there will most likely be is some increased fees to their accountants as the format of limited company accounts needs to be correct and probably means a higher level of staff will be involved in it.