ChipFlip
Intel's naming change makes this comparison harder than it should be. The Core Ultra 5 245KF sells for $200 used, the older Core i5-13600K for $220. Newer chip, lower price, different socket. Working out which is the better buy takes a minute.
The socket difference is the whole story
The 245KF uses LGA1851, Intel's current socket. The 13600K uses LGA1700, which Intel has finished with. If you want any chance of a CPU upgrade without a new motherboard, only one of these offers it.
Whether that matters depends on how you buy. Plenty of people run a CPU until the whole machine gets replaced, and for them socket longevity is worth nothing.
Gaming performance goes to the older chip
This is the uncomfortable part of Intel's Core Ultra launch: in gaming, the 245KF often trails the 13600K it nominally replaces. Arrow Lake moved memory and interconnect around in ways that hurt latency-sensitive workloads, and games are latency-sensitive workloads. Reviews at launch were blunt about it.
So you'd be paying $20 less for a newer chip that's slower at the thing most buyers care about.
Efficiency and productivity favor the 245KF
Arrow Lake draws considerably less power than 13th and 14th gen did, and it runs cooler. In multithreaded productivity the two trade blows, with the 245KF ahead in some workloads. It also sidesteps the voltage instability issues that affected 13th and 14th gen parts, which is a real consideration when buying used.
The KF suffix means no integrated graphics, same as the 13600K's K counterpart would have. Both need a discrete card.
Bottom line
For a gaming-first build, the 13600K is faster and worth the extra $20, provided you confirm it's running updated microcode. Pick the 245KF if you want lower power draw, a socket with a future, and no exposure to the 13th and 14th gen instability history.