C19 is a higher temperature rating (16A/240v). My Shunyata units had C19 sockets to connect to the wall socket. We have fused 13A sockets and C15 are safe, but could warm up. The Everest is rated for 16A (UK) and 20A (US) output. If you have a component drawing more than 10A (perhaps 12A or 13A in the US?), C19 might be preferable.
At the end of the day, these plugs have IEC ratings and you just need to know the maximum current draw of the appliance and then use the appropriate plug. If it's supplied with C15, then C15 should be safe. Maybe that's wishful thinking. My most powerful component has an 8AT fuse on the power inlet, will blow almost immediately the draw exceeds 8A, so C15 is always going to be safe, my Puritan conditioner is limited to 10A per socket, but also uses a C19 outlet to the wall so clearly is designed to draw more than 10A. A 13A fuse will not blow even with a constant current draw of 20A.
So on second thoughts you should also look at the fuse protection of the component as the fuse may blow long before a C15 plug overheats.