I think the key is to discover your bottlenecks.
Are you bottlenecked by a single processing thread? If you're on a single thread both CPUs have a similar clock speed, and any cores beyond the first won't help. If you are CPU limited then this could be a big part.
What else is going on in other processes? That older system doesn't have hyperthreading (it is a single processor only), so if your newer system has more background stuff going on that uses all the cores then hyperthreading may be harming the process of a single-threaded program. Again if you are CPU limited this can be a factor. Single CPU means no tests for cache consistency between processors, which could account for a tiny speed difference.
What about bus traffic on the motherboard? The older single-core machine is less likely to saturate the hardware bus if that is your bottleneck, and will similarly have less memory contention if those are your bottlenecks.
It probably isn't in issue, but on much older games and software there were big differences that are hard for newer machines to handle. For example, if the game were a DX5-era system and one system was a DX5-era card (it isn't in this case) then the older architecture may perform better even though newer cards can handle newer processes faster. This was also true when older graphics cards were focused on tiles and sprites but newer cards were focused on textures and meshes. Today's cards handle point clouds amazingly well, but need to work hard to emulate old APIs. If you are using extensive 2D APIs then perhaps the newer cards don't have a good hardware-accelerated way to implement the operations.
It could be memory speed where the program has tons of cache misses, and both have similar speeds for accessing main memory.
It could be disk speed where the program is spending most of its time waiting for disk reads, and both have similar speed hard drives.
It could be something else entirely, those are just guesses without seeing the systems in motion or seeing any measurements.
Figure out what the bottlenecks are. You'll likely find they're the same or similar bottleneck on both systems.