I am misunderstood when I say can't--almost as often as I'm misunderstood when I do my modern dance interpretation of Aspects of the Theory of Syntax. Many BrE speakers have difficulty hearing the difference between most AmE pronunciations of can and can't.
In both dialects, it's not really the 't' that helps us tell the difference between the positive and negative words--that sound is mostly swallowed at the end of the word. It's the vowel that makes the difference. In standard BrE, the vowel quality is the clincher. Can rhymes with pan but can't is pronounced like the AmE pronunciation of Kant (see the comments for more discussion). In other words, the vowel in can't is considerably further back in the mouth than the vowel in can. (One feels the need to mention the old chestnut: Kubla Khan, but Immanuel Kant--but note that Khan and can are pronounced differently.)
In AmE, the vowel quality is very similar between the two forms, but the length of the vowel in can't is shorter. (By a 'shorter' vowel, I literally mean 'shorter'--i.e. not pronounced for as long.) I explained that fact to my students yesterday, and once they knew that they got much better in the "which one am I saying?" test.
Pronouncing can't the other way is a favo(u)rite way for British singers to make themselves sound more American, and for American singers to make themselves sound more British. (I suppose either identiy has some cachet, depending on what kind of sound you're going for.) I've got to run to London to celebrate some fellow Librans (yeah for us!), so will leave it as your homework assignment to identify one example of each!