I have done a clean install but I was wondering if it would make any difference if I just reinstall all my old applications or if I just move my backup applications folder.
It really depends on the application, so there is no correct a) or b) solution that would encompass all your apps.
Many Mac apps are 'installed' by the simple expedient of copying the app from a disk image to the Applications folder.
So long as your new install uses the same Apple ID as your old &/or you have any required activation keys, all is good.
The issue arises with apps that needed to run as an Installer, asking permissions to modify System items, install background processes etc. For those you will need to correctly re-install.
I guess empirically, the 'suck it & see' method might be the simplest...
...copy the Applications folder across, see which ones work... [avoid copying any default Apple Apps]