How to back up the hard drive to another computer?

Not sure I follow you here... You took the HD out of your G4 system and want to copy off the apps & data to your new system is that correct?

So what is your new system, another G4 based system?

And was your PowerBook G4 still working?

If you are trying copy things to a newer Intel based system you'll have issues with the Apps as they likely won't run on the newer system depending on the newer systems OS version (Lion onward).

As for your pictures and other data you can copy them off but using a IDE to USB HD adapter. Here you plug the HD into the adapter then into your system. You may need an external power source to help power the HD for the task. If the drive does not show up under the finder its possible the drive is dead.

Anther option here is putting the drive back into your still working PowerBook system and then setup your PowerBook in Target Mode. This allows you to use the system like the adapter then you can connect your two systems via a FireWire cable to copy the files.

Lastly, if your system is still running but you don't have a FireWire cable or your new system does not have FireWire, you can also setup your PowerBook as a shared resource on a network so via Ethernet you can access the HD from a second system to copy the needed files.