But if coral is so vulnerable to warming, why is it still here after hundreds of millions of years?

Fair Home » Tropical Philippines at Great Risk From Melting Ice

According to Comiso these reefs could face the brunt of the damage from climate change, as they are highly vulnerable to shifting sea temperatures.

Even minor temperature shifts cause a phenomena known as “bleaching” which means that the coral looses it vibrant colours and becomes pure white or bleached.
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Coral is intolerant to this warming and struggles to deal with the change.

coral: Definition from Answers.com

Although corals first appeared in the Cambrian period,[14] some 542 million years ago, fossils are extremely rare until the Ordovician period, 100 million years later, when Rugose and Tabulate corals became widespread.