Can philosotards explain this?

But OP seemingly was about:

Yet in rhetoric bringing up a specific counter-example is considered a logical fallacy.

A specific counter example

It depends on the context. A single counter-example only disproves a generalized statement if it is postulated as an absolute truth. But this is not always the case with generalizations.

If I say "women are physically weaker than men" I can mean either a) that all women are physically weaker than all men, or b) than women are generally, but not necessarily always, physically weaker than men.

Mathematics and the more exact sciences (e.g. physics and chemistry) are concerned pretty much exclusively with truths of the former kind. In "softer" sciences, where you can't have the same kind of exactness due to a combination of methodological difficulties and the higher complexity of the systems being studied (e.g. biology, medical science, psychology, sociology etc.) you have to make do with truths of the latter kind (e.g. "people with a very high BMI are generally sicker and more short-lived than people with a normal BMI," statements of this kind frequently being put, for reasons of convenience, in simpler but not entirely accurate terms, e.g. "fat people are unhealthy").

So if it is specified whether the claim is exactly absolute or if the claim is general then a specific counter example could not be considered a logical fallacy?

Pretty much. If someone says "every single immigrant is a criminal," then it's obviously not fallacious to counter that with a specific example of an immigrant who isn't a criminal.

Of course, people usually aren't that exact in debates, because debating and rhetoric is less about truth than about the appearance of truth. If you want to be as convincing as possible it's often to your advantage to avoid exactness and to make use of every fallacious argument that you can, as long as those arguments have the air of truth about them and are unlikely to be effectively countered by your opponent.

>because debating and rhetoric is less about truth than about the appearance of truth.
>make use of every fallacious argument that you can, as long as those arguments have the air of truth about them and are unlikely to be effectively countered by your opponent.

With the internet existing and all, why would ^^^ even exist?

>In math and science all one has to do to disprove a theory is to show a single counter-example.
Science don't actually work like that, user. There is a very narrow understanding of it.