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On Wednesday, the Obama administration launched its new counter-terrorism strategy [2] - the first published revision since the Bush presidency. It did so from a position of strength, having finally tracked down and killed Osama bin Laden. Recent polls show rising approval of Obama's handling of terrorism, boosting his overall ratings and helping him hold the generals to the deal he struck in late 2009, to start bringing troops home from Afghanistan in big numbers.

The new strategy contains a fairly detailed discussion of the Arab Spring, arguing for applying "targeted force on Al Qaida at a time when its ideology is under extreme pressure" from events in North Africa and the Middle East. By contrast, Britain's revised Prevent strategy [3] published three weeks ago, mentions these events only once - in a footnote, saying with characteristic bureaucratic obtuseness that it's too early to say what the effects will be. The British system has probably been too busy thrashing around on the argument [4] which has divided our politicians, counter-terrorist officials and experts over the last year - the argument about whether the real enemy is terrorism, violent extremism, or extremism more widely.

That question was supposedly resolved in the Prevent strategy, which committed the Government to broadening out its counter-terrorist efforts to include non-violent extremists, defined as people and organisations who disagree with our "core values", including democracy, equality before the law, and universal human rights. David Cameron had flagged this shift in his Munich speech [5], when he explicitly criticised the previous governmentfor focusing too narrowly on terrorism and violent extremism, and failing to be sufficiently 'muscular' in standing up for our values.

The most interesting thing about the new US strategy from a British perspective is that it essentially takes the diametrically opposite path [6]. President Obama's foreword asserts that "we must define who we are fighting with precision and clarity", and the strategy states that "by ensuring that counter-terrorist policies and tools are narrowly tailored, and applied to achieve specific, concrete security gains, the US will optimise its security and protect the liberties of its citizens."

At the time the Prevent strategy was published I criticised [7] it for being sloppily written and not properly thought through - as well as representing a fundamental wrong turn. It fails to make clear whether the Government believes in tackling non-violent extremism as a matter of principle, or because it thinks this will reduce the risk of terrorism. (I noted that either or both are legitimate - if arguable - positions, but the Government needs to be much clearer which is driving policy in which area, as they can have quite different implications.) The instruction to universities and those who oversee internet provision in any "public institution" to intervene directly against groups or individuals who "do not share our core values" is both confusing and dangerous. The wording slides between asking universities to monitor and take action against people or groups involved with terrorism - which academics may tolerate - and asking them to monitor and take action against groups who disagree with our "core values" - which they surely cannot. It slides between talking about blocking online content [8] which is "unlawful" - which is a matter of fact, and relatively uncontroversial - and blocking online content which is "harmful" - which is a matter of judgment, and highly controversial.

There are very specific risks here for freedom of speech and thought, on campus and online, which need to be addressed urgently. The deeper question is whether Britain is going in the wrong direction on the fundamental question of how we win the argument for our values - as well as whether it makes sense to go in the opposite direction to the US, given how closely the two countries work together.

Cameron was praised by the likes of Matthew d'Ancona [9] for characterising the struggle against the ideology behind terrorism as a "new Cold War". The analogy was hardly new - Gordon Brown was using it five years ago [10] - but more important than who got there first, is which of the two has understood the analogy in the right way. When Brown invoked it, he remembered how:

It was fought not only with weapons and intelligence but through newspapers, journals, culture, the arts, literature. It was fought not just through governments but through foundations, trusts, civil society and civic organisations. We talked of a cultural Cold War - a war of ideas and values - and one which the best ideas and values eventually triumphed. And it is by power of argument, by debate and by dialogue that we will, in the long term, expose and defeat this threat.

The new US strategy emphasises a similarly positive approach, relying on the power of American values to win arguments and attract admirers, rather than the negative approach of attacking, or trying to silence, those who hold different values. The key, the strategy says, is winning the positive argument, engaging with and discrediting the ideology of Al Qaida and its supporters and adherents, limiting its resonance by addressing the grievances it feeds off - and avoiding doing anything that will discredit ourselves and undermine our values in the eyes of others.

Brown struggled to put his own version of this strategy into practice, and we will have to wait and see how the Obama administration fares. Some will observe that they need to do better in particular at avoiding undermining themselves: most notoriously in the continuing Guantanamo saga, but also in their response to the Wikileaks episode (they would have benefited from reading the section of the new strategy headed 'Balancing Security and Transparency'). But the strategy is the right one, and it contrasts favourably with Cameron's Munich speech, with its unfortunate echoes -"we need to wake up to what is happening in our own countries" - of the McCarthyite paranoia and conspiracy theories which are a less proud part of the Cold War story.

Cameron likes to sum up his new approach as 'muscular liberalism', but the Munich speech shaded from muscular into macho, and the Prevent strategy verges on the ham-fisted. Obama's alternative points us to a better alternative: confident rather than muscular, more nuanced - but also more open and more positive - and in the end more likely to succeed.

Matt Cavanagh is Associate Director at the Institute for Public Policy Research (IPPR)