Having checked the DXDIAG files, both computers meet the minimum hardware requirements for the game, and the desktop has reasonably up-to-date software. You might want to install newer drivers for the graphics card, which you can download from here:
Download the English (US) GeForce Game Ready Driver for Windows 7 64-bit, Windows 8.1 64-bit, Windows 8 64-bit systems. Released 2018.3.27
However, I suspect that this won't help with the error the desktop is experiencing (although it doesn't hurt to try it - just download them, run the file, click OK on each step, don't change anything, and let them install. Restart the computer once its finished).
The actual error involves a problem with the game trying to run a Python installer (Python is a computer language, and probably the one used to make the game). I recommend uninstalling the game, restart the computer, and then download the latest version of the game from here:
Toontown Rewritten is a free-to-play revival of Disney's Toontown Online. Create your own Toon and join the battle to save Toontown from the evil robot Cogs. Play now for FREE!