"and what difference does a politicians sexuality make to his/her ability to do the job?"
This is a valid question, so just thought I'd comment on it.
By and large, you're right. A person's sexual orientation does not (or rather, should not) affect their job. What Livingstone was trying to do, however, was refer to cases where it does matter (albeit he did so in a rather stupid way): Cases where people in positions of power will champion 'family values' or oppose gay rights legislation, while not doing so themselves. Who a politician like that sleeps with is just about the only case where it is relevant and does reflect on their job.
A good example of the same thing, I think, is Newt Gingrich. Ordinarily, it wouldn't really matter with regard to him doing his job if he was sleeping with someone young enough to be his daughter and lying about it. But when he was ferociously attacking Clinton in the 1990s, and at the same time attempting to champion 'family values' more generally - it was something of immediate and important relevance because it reflected on the man's integrity and trustworthiness.