The host hardware should have no influence on the emulated game, other than the speed of the emulation. That's what emulators are for after all (; .
Here's some ideas:
Is the host CD drive letter the same on both machines? You could try creating an image of the CD (much more convenient anyway) and mounting it with the "imgmount" command.
Have you double-checked the mountings and made sure all files are available to DOSBox when the emulation starts? To do this "rem" the lines that will auto-start the game, start DOSBox and issue "dir" commands for c: and d: .
If you've copied the already installed files from the "old" machine to the "new" one: are you sure that the game data on both machines is identical? This can be checked with a "file comparing" tool, Total Commander i.e. has the functionality built in. You should byte-compare each file to make sure everything is there.
Are you using the exact same DOSBox version on both machines?
You could also try different "output" values in dosbox.conf to see if the new system has problems with certain outputs. That's probably not the case though.