There is no difference. A contractor has to have a valid license (except in limited circumstances) to perform construction work. A contractor that has a contractor's license is "duly licensed."
Having a valid license has nothing to do with a website - but you can find out whether a contractor has a valid license by checking the CSLB website.
The CSLB will issue a certificate for hanging on the wall to a licensed contractor and sends contractors a license card designed to fit into a wallet.
I think they are one and the same. It seems like a distinction without a difference, or mere semantics.
For confirmation of your hunch on this, I would speak with licensing attorney Christine McCall.