Yeah, clay and paint cleaners (especially chemical paint cleaners) work differently and can have different results. Both work differently from abrasive polishing too. All can have their place.
I really don't use paint cleaners all that often, although there are a few (e.g., Pinnacle PCL) that I do really like.
Paint cleaners are especially good when you have weird staining and/or oxidation or when you want to strip off an existing LSP (in a controlled manner, as opposed to doing a decon. type wash) in order to try something else.
A good, cheap, near-functionally-nonabrasive paint cleaner that's OTC is Meguiar's Deep Crystal step #1.
Some cleaner waxes (e.g., * IIRC* Meguair's Color-X) have pretty good chemical cleaners in them.