You get better/faster/more thorough cleaning with a dedicated gun cleaning solvent.
The solvent attacks and removes all fouling like copper, carbon, and powder fouling, as well as dirty lubricant.
Few CLP's can clean copper fouling.
The CLP type products can be used as a do-it-all field product which allows carrying one bottle that can do a reasonable job of cleaning and lubricating.
Another factor is that the cleaner part of the product will keep fouling in the action soft and allow the moving parts to sweep the fouling out of the way and allow the weapon to continue to function longer.
A dedicated cleaner just does a faster, better job when what you want is to CLEAN the gun.
I recommend using a dedicated cleaner to clean, and a CLP to lubricate and protect. As an added benefit, the C part of CLP will keep the weapon working and make cleaning easier when you do clean it.
In short, when what you want to do is clean a gun, use a cleaner.