Or in your case, copy the complete folder, go to the new folder, replace the linked kicad_pcb file (in the new folder) with the modified one. And then open the project in the new folder.**
Alternatively, you could move the non-modified pcb to a new name and create a (soft) link to the new pcb with the same name of the old pcb. (not a recommended way)
AFAIK you can not have a PCB with a different name than the project (without any other tricks like a link).
The other, much better, method is to use a version control system, which is a good idea in many other regards too. Then you could save, create a new branch, save it there and if you want to go back to the master branch that doesn’t have the modifications.
** On Linux that would be:
cd /path/to/pcb/project/..
cp -r project project_new
cd project_new
mv modified_pcb.kicad_pcb projectname.kicad_pcb
kicad ./projectname.kicad_pro