Your chances of getting anything to move just plugging in a port from a PC are about equal to using your car as a space shuttle. I suspect that based on all the TTL chips and ROM/processor chips that the machine movements are through firmware on the card and not by outside step & dir control. It's tempting to try and use an existing driver setup (I have dozens of machines that I have bought and stripped for parts over the years) but since the machine are design for aspecific purpose often the drivers fall short of what is needed for CNC machines. Typically the L298 bridge drivers (the multi-leg chips on the heatsink) use L297 driver chips to get the phasing right using single ended step switches. There are none on your design. Based on the date codes on the chips it would appear the machine was built sometime after 1997 (but not LONG after!). Motion technology (and expense) has changed a lot in the last 12 years.
Rejoice in the fact you have some expensive mechanics to marvel over and perhaps adapt to a design.
If your background is not electronics then digging into a custom design to try make it a "Silk Purse" will be an exercise in frustration. It's possible the designers did some very sophisticated motion logic and used microstepping or even wave type motion (ROM table lookup??) to get precise motion. If you had the schematics and the help of an EE it might be possible to build an interface that could make it work with conventional low cost CNC interface options (aka PC control) Even then, you can duplicate the entire drive and logic for CNC for $299.00 with a G540 Gecko.