If you say that you know David Cameron is prime minister or you know today is Monday, by using the word 'know' you imply that the assertion which follows it is true. The verb 'to know' is called factive for this reason. Should what you claim to know turn out not to be the case - as in 'I know that Margaret Thatcher's first general election victory occurred in 1977' - then the knowledge you thought you had wasn't, in the event, really knowledge.
Contrast that with the verb 'to believe'. Saying you believe something to be the case doesn't presuppose that the belief is true, even when you think the belief is true. To say 'I believe that Margaret Thatcher's first general election victory occurred in 1977' would be evidence that you have a false belief in this matter (since the relevant date is 1979), but it wouldn't show that you didn't believe what you'd claimed to believe, in the way that in the previous case your claim to have known something, was overturned by that something not as a matter of fact having been a fact.
OK, so how about statements to the effect that someone remembers something? For example, I remember that I went for a jog on 5 July 1980 during the Borg-McEnroe Wimbledon final - in the hope of inducing a victory for Borg. This looks, at first, more similar to knowing than to believing, since for me to have remembered what I think I do, Borg and McEnroe must have played in the final on that date. As it happens, I can check that they did, both in a running notebook I used to keep and by reference to other sources. On the other hand, people often misremember things, and when they do, we wouldn't necessarily want to say they didn't have the memory they claimed, only that it turns out to have been a false one. In this a memory is more like a belief, even when it looks like implying a claim to knowledge.
In yesterday's Sunday Times, John Carey reviews The New Science of Memory by Charles Fernyhough, which covers, among other things, how fallible people's memories are. Carey writes (£):
His [Fernyhough's] main point is that memories are not stable possessions stored in some internal library. What they are remains a mystery, but current thinking is that they are changeable imaginative constructs that, in order to meet the needs of the present moment, rearrange and adapt traces of past happenings that have lodged in our brains. Or something like that. Whether this definition is true or not, it is certain that memories are often false... and this applies even in cases where we might expect accuracy. The famous "Proustian" effect, when a taste or a smell brings back floods of memories, would, you'd think, be dependable... But experiment shows that Proustian memories are as likely to be false as any others...
For most people memories are deeply important. They make us who we are. So we are alarmed when we are told they are just a mass of fictions. Fernyhough argues that we should, rather, rejoice in our own creativity. The fact that our memories are false shows that "we are natural-born story tellers". Memory errors are really "marks of success".
Carey has his reservations about this, and so do I. There's nothing wrong in itself with celebrating memory's story-telling and identity-defining functions, its creativity and so forth. But if one does that one needs to
remember keep firmly in mind the idea of memory as dealing in 'imaginative constructs', and be more cautious than one otherwise might of its would-be knowledge-bearing function. This would accentuate the need for corroborative memories when one is considering the evidence of witnesses (in law, in historiography, etc); and also for a degree of caution in day-to-day life when individuals simply assume their own memories are authoritative in an informational sense. They might be or they might not.