Many people would say that if you realise that you are in the wrong, then you should always apologise. Perhaps, they'd exclude sociopathic situations where this would be used to manipulate you, but that'd be it.
However, it's easy to forget that apologising creates a cost for the person who is apologised to. They have to read your message and perhaps write a reply. This later component is tricky if they aren't convinced that you've made up for it. It reminds them of an experience they might want to forget. Further, it requires them to deal with a topic they may be completely sick and tired of.
If you apologise, it should be because it helps prevent or mend a rift with the other person. You should be extremely cautious about apologising when it's because that's what you think a nice person would do, as opposed to something more specific, since those are precisely the situations where you are likely to end up apologising with no benefit to anyone.
Now most people don't apologise enough and so this is probably the wrong advice for them. But Less Wrong samples a particular segment of the population and I suspect it includes a disproportionate number of those people who over-apologise.