This has a name, "Tu Quoque" and it is, in fact, a logical fallacy.

If someone calls you out for kicking puppies, it is not a fair counterpoint to say that they kick puppies too. Their actions are immaterial to your actions. If the question is one of morality, your actions are immoral regardless of whether or not your accuser is similarly immoral.

It's considered a formal fallacy. That doesn't mean it's necessarily a bad point in context, it just isn't the point it's often presented as. If someone is making a long speech about how X is bad, when they do it themselves all the time, then you're probably right to discount their arguments - if they're not stupid, they're dishonest. But pointing out the hypocrisy is technically "off topic" if you're arguing whether X is actually bad.

Very often it's not exactly the same criticism, and is just deflection - they are hoping to start arguing about whether the accuser's actions are equivalent, rather than whether their actions are objectively bad.

For example: A accuses B of allowing poor people to starve to death and B replies that if A cares so much about poor people, why does he want to put taxes up?

Pointing out hypocrisy can be relevant, but often is just whataboutism. It is perfectly valid to have different opinions on similar situations, because no two situations are completely equivalent.

That said, anyone arguing in good faith should be prepared to examine their own biases as well - as long as they really are relevant to the latter at hand.

I see a few different ways this could play out, and it highly depends on the specifics of the point being debated. Fundamentally though, pointing out hypocrisy is a good way to criticise the speaker, but it does nothing to refute the point that they are making. It's a form of ad hominem argument—a term which, contrary to how it's sometimes used, doesn't just mean "insulting your interlocutor", it specifically means that your argument doesn't address their point, but merely addresses the person making it.

It could be that they simply didn't realise they're being hypocritical. You explaining it might make them realise the breadth of the argument they were making and cause them to reevaluate and realise that the implications of what they were saying were more severe than they previously thought. You haven't actually addressed their argument, but you didn't need to. Your goal was to help them realise for themselves that it was wrong. They then update their belief in line with this. (Which, depending on the specifics of the argument, and whether or not they and you are both acting in good faith, might mean the end up completely abandoning the point, or simply redrawing the precise boundaries of the point, or something else.)

It could be that they realise, and it's something they either don't care about, or do care about but don't see it as refuting their argument. For example, in an argument about addiction, an addict might recognise their addiction and be working on overcoming it, or just be living with it. You pointing out that they're addicted to [whatever it is] doesn't really advance the argument at all. Either they're trying to work on it, or maybe they've acknowledged their addiction but don't really care. Whatever the case may be, how sound their point is is not affected by this quality of them.

The correct thing to do is to report to mods and leave.