At the moment bowls and mugs don't have that much use - I didn't have enough time to fully implement the functionality before hitting the deadline for Steam release.
The plan is that jars (and wooden containers, to be added) should affect the storage of crops.
I'm not yet entirely sure what would be the actual impact of mugs and bowls; probably something like not having them would make their life more unpleasant. (There is a hidden variable simulating how satisfied they are with their lives, that has a subtle effect on things like recovering from ailments etc)
Also, high quality clayware will have good barter value. But this is also something not yet fully implemented.
What comes to cauldrons, they do have a real impact on the simulation. Lacking a cauldron (or, having too small a cauldron relative to mouths to feed) means that all cooking must be roasting or other such methods, so cooking takes longer. With a proper cauldron it is easier and less time consuming to prepare stews to feed the family. If the cooking is being hampered due to lack of a cauldron, I think there is some feedback about that in the info for cooking routine - but sure, the UI could be more intuitive, so I still need to think about ways to make the relevant information easier to spot, without constantly spamming too much pop-up dialogs all the time everywhere...