When Gordon Brown resigned two months ago, he left behind no thank-you notes for the staff at Chequers who had looked after him so often during his three years in office. After David Cameron spent his first weekend at his new weekend residence, by contrast, he returned to London leaving a handwritten note of appreciation.

This anecdote does not tell you everything you need to know about why, after a mere eight weeks, Cameron is already proving to be a much better prime minister than Brown ever was. Nor does the claim, which I have also heard but in this case not confirmed, that the Chequers staff learned to keep the best china under lock and key during Brown's visits because of their fears that he would break it in his rages, prove it either.

But these tales suggest why Cameron, in such a short time, has already become such an asset to his party, to his government and, if one may say this, to his country, too. Good manners should be a necessary not a sufficient condition for advancement in politics – as the infinitely well-mannered Michael Gove has discovered this week over school building projects – but they certainly help to turn away wrath at a time of upheaval.

Like Tony Blair before him, Cameron deploys his courtesy and charm for political purposes. Watch him in Question Time as new Labour MPs ask him barbed questions. Cameron does not sneer or talk down to backbenchers, as Brown did. Instead he flatters them. When Labour's Kate Green asked her first question to Cameron he began his answer by praising her "incredible work" as head of the Child Poverty Action Group. This week, when Gloria de Piero's turn came, Cameron complimented her for leaving the "warmth of the GMTV sofa" to go into politics.

If you are tempted to think this sort of courtesy across the political divide is mere bourgeois triviality, remember how effective this maturity of style has also been in larger contexts. In the last two months, nothing has become the prime minister more than his Commons statement of regret for the Bloody Sunday killings. The speech was a model, and when Cameron said, "On behalf of our country I am deeply sorry", the applause outside Derry Guildhall almost seemed to wash away 40 years of hurt.

A certain grace has marked Cameron's outings on the international stage, too. He has handled European meetings with a deftness and in a tone which have surprised those who feared an immediate lurch to the right. You may counter that there have been missed opportunities, or that he was a key player in letting too much of the air out of the G20 – a forum in which Brown's big thinking was missed – but at least he did not talk up his role in Toronto for the benefit of the UK media, as his predecessors would have done. Cameron has sensibly lowered the bar for British self-esteem in the world – and a good thing, too. He approaches his visit to Washington later this month with fewer political hang-ups than any of his recent predecessors.

On the most important international issue of all for Britain, Cameron could hardly have been clearer. He wants out of Afghanistan. Exactly how it will happen remains vague, but the intention is clear. Invited by Labour to say he intends to stay in Afghanistan until the job is done, Cameron sharply demurred. We will not be in Afghanistan in 2015 – the date of the scheduled next general election – he said on Wednesday. Cameron may have no clear exit strategy, but his strategy is exit all the same.

At home, the relationship with George Osborne has always been at the core of Cameron's strategy and it will be the economy and the fiscal tightening that will make or destroy both men's reputations. Yet it is the new bond with Nick Clegg that reveals most about the Tory leader. In the eight weeks of the coalition, Cameron has consistently gone out of his way to deliver for the Lib Dems – in the coalition programme, round the cabinet table, in the budget (though not enough) and over electoral reform. He probably needs to do more. Any Tory leader would have had to do a deal with the Lib Dems. No alternative leader could have done it as well.

Cameron has not just taken to the realities of coalition better than any other Tory. He has also done it infinitely less condescendingly than Brown or any Labour leader would have done. He recognises that he is delivering a deal, not a sell-out. Yet in doing so, he has pulled the Tory party further towards both the centre ground and an acceptance of coalition politics – and pushed Labour off both – than many would have believed possible. His seizure of the opportunities of 6 May to put liberal conservatism at the heart of this government's project has been audacious. Its long-term impact on the Conservative party is not yet clear. Its short-term impact is immense and, in Tory terms, almost wholly desirable. The old right (like the old left) can only gawp and grumble.

The imperatives of coalition have helped cement Cameron's authority in ways that not even he can have foreseen. Like Blair, Cameron came to power outside parliament rather than within it, by climbing the outside of the building, as Bill Rodgers said of Blair. Before the election he was criticised within his own party for ruling through a handpicked oligarchy, again like Blair. The election has changed all that. Coalition requires Cameron to consult, make deals, and actively manage both his party and his government. Necessity has demanded the return of proper cabinet government. The seductions of cliquism and presidentialism to which Blair succumbed are off limits because of the hung parliament. A good thing, too, and Cameron has adapted with an admirably sure touch.

The polls show what a key asset Cameron is to the coalition. The government's approval rating is +8 points, according to this week's YouGov/Sunday Times poll. But Cameron's own rating is +28, with almost unanimous approval from Tory voters. Take Cameron out of the equation, and the coalition's standing would be fragile indeed.

These are still very early days. The coalition has to get through difficult votes on AV and negotiate the most difficult spending round in a generation. The economy may tank. Yet in these first weeks even opponents should concede that Cameron has played a blinder. He is showing himself as potentially the best all-round prime minister of the modern era. Labour's hopefuls should learn from him. No doubt about it, Cameron wins this season's political golden boot.