I lost my T42 and I'd like to buy a used one on e-Bay and copy my software and files onto it from a back-up drive so I don't have to re-load everything. I had a 2373-CD8 and I was wondering if I have to have the exact same motherboard for the back-up copy (OS, apps & files) to work?

No, in fact you can copy the T42 image to a T40, T41, T42, or R50, R51 and it will work.

Any T41, T42, R50 or R51 will do.
Your model had the ATI Radeon GPU.
In view of problems with these ATI GPUs, you may be better advised to look for one of those models with an Intel GPU.
In any of those above, you can insert your hard disk and be up and running. Only differences may be the wifi card drivers and (if it's an Intel GPU) the graphics drivers.