Hi BP61,
If you open File Explorer then navigate your way to C:\Program Files\Tinytown Rewritten folder do you see a folder inside called Logs? If this is a 32-bit application it may be C:\Program Files (x86)\Tinytown Rewritten.
If you do open that Logs folder and then inside open any log file present. Take a look at the log to see whether there is a specific error recorded.
It could be something as simple as your OpenGL driver not supporting Hardware Acceleration in which case you could probably update this driver via Device manager, or physically go to your Manufacturer website or the Manufacturer of your Graphics card's website and update your driver.
You mentioned it works find on your laptop, so you could compare Device Manager driver settings on this with your Desktop (noting of course each machine may have a different Graphics Card chipset, but if both machines are made by same company then it might help to compare the two).
Regards,
Regedit32