I should have fact checked that small detail before typing a response on my phone...I believe the my reply gave insight to OP his comment indicated he wasn't aware of, despite my small error.In the US, you only need a CITES export permit for species that are CITES II listed (corals and seahorses for example). All non-listed species just need to be shown on a USFWS 3-177 import permit.
The export of any CITES listed species from any country requires a permit. This means unless the aquarium coral (or fish, if listed) is from the USA, an export permit is required by the exporting country for shipment. The import permit is dependent on the country and Appendix the species is listed in, if listed. (All stony coral species are Appendix 2). It isn't everything that requires a permit and I dont really want to spend my time reviewing the list of species that do require one, but as you mentioned, there are still permits required since the USA is a member and carries out the provisions set by CITES.
CITES sets the provisions.
I responded to a comment by OP stating he wasn't aware of organizations or agencies regulating the taking or trade in species outside the US and mentioning CITES informed him of am international agreement that does just that.