If the reason for your post was knowing the difference between those options, the reply would be:
"Restart Couatl" is commonly used when you want to restart all services immediately, regardless of what they were doing, and start from scratch, without waiting for the services being completed.
"Restart Couatl and rebuild the airport cache " does the same, and it also forces reading again your installed airports. It's commonly used because with some 3rd party sceneries that use external configuration tool that alternate between different AFCAD files, depending on the method they use to switch between them ( copy or rename ), GSX might not always able to automatically detect a change, but it will surely will after a Cache rebuild.
That's why there are two options.
Also, you description of the menus is not accurate, since both "Restart Couatl" and "Restart Couatl and rebuild airport cache" are under the "Couatl settings" menu, together with "Key Mappings" and "Show YouControl Menu", while the "Live Update" menu contains "Check Live Updates".