You need to invest in a $20 multimeter tool in order to verify which wires are live and which breakers (if any) that they are connected to. You may think you know it is on the 60A breaker, however that is not sufficient.
First you uncap the red and white wires. Check the voltage between them and each other, between them and the bare (ground) wire. Should be 240v and 120v.
Now, back to speculation, it appears that you have two sets of wires coming from your breaker panel. One is large, the other is small. The large wires have been disconnected from their outputs, the small wires have not.
A 60A breaker is too large for a 14-50, so you will either need to replace it with a smaller breaker or install the wall connector. The wall connector is for outdoor use, so that is definitely what you ought to be using.
It is also very possible that the large guage wires were never in-use. 60a is way too much power for a hot-tub, in which case the hot tub uses the small wires, and the large wires were run by someone with foresight of a future need. In that case, you should try to preserve the hot tub wires, as it looks like a professional job.
I don't see any need for a subpanel here.