64-bit desktop ISO vs 32-bit desktop ISO: EFI vs legacy boot mode
Solution 1:
The 32-bit Ubuntu is BIOS (legacy) only, whereas the 64-bit Ubuntu supports both BIOS and EFI boot modes. Thus, the 64-bit image should work on your older computer, provided that computer has a 64-bit CPU. The x86-64 (aka AMD64, x64, or EM64T) architecture dates to 2003 (with AMD's Opteron line), but it started to become popular a couple of years after that. Even today, some computers still use 32-bit CPUs, although they're mostly low-end or embedded devices. Thus, without further details, it's unclear whether your older computer is 32- or 64-bit.
The 32-bit Ubuntu should run on 64-bit computers, including most EFI-based systems; however, to boot on an EFI-based computer, a BIOS-based OS (including standard 32-bit Ubuntu live CD/USB images) requires that the EFI have a feature known as the Compatibility Support Module (CSM), which is a BIOS-compatibility layer. Most modern EFIs have this feature, but it's often disabled by default, so you may need to enter your firmware setup utility to enable the CSM (aka "legacy mode" or "BIOS mode"). The details of how to do this vary wildly from one EFI to another, so it's impossible to give simple and accurate instructions for doing so that apply to all computers.
Solution 2:
The 64-bit Ubuntu live USB disk will work even if you turn off UEFI. The 32-bit version of Ubuntu does not support UEFI. So, if you are not having trouble getting the older PC to work with the 32 bit version of the Ubuntu Live USB, you can get both computers to run it as long as you turn off UEFI on the newer machine.