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I think you basically answered your own question, the difference is that humans are aware of themselves and aware of context and impacts of their actions.

Saying the climate change is natural because humans are natural may be true in a sense, but it really isn't meaningful, it's just playing with words. Thermonuclear winter would also natural in that sense.

The more meaningful discussion is a) what's happening, b) what impact would it have, and c) what can/could/should we do about it.

If (for example) we could see the future and our actions precipitated a mass extinction of sea life that was the basis of the food chain, resulting in the loss of 90% of species or 90% of humans or something like that (not saying that's what will happen, just picking something obviously extreme for illustration), on one hand you can say that's a natural event because humans are natural and that earth has survived such things before (or even that earth becoming lifeless as a result is still natural), but that doesn't answer the question of if such an outcome is something that we should allow to happen (being in the position to be able to do something about it).

Then it just becomes a question of weighing potential consequences against values and costs of actions...