As others have mentioned, it depends on your dental plan and whether or not your dentist is a participating provider.
The plan is the insurance and provider is the dentist.
Looking at the picture and generally speaking, the "claim amount" is what your provider charges anyone who walks in off the street.
A participating provider already has an agreement with the plan to charge a certain "flat rate" for covered procedures.
The amount "paid by plan" is generally this "flat rate" negotiated rate that the plan has agreed to pay their providers - whether they negotiated it or not.
The amount "you may pay" is the difference between the two that your provider may or may not bill you for.
My dentist is not a participating provider with my current plan. However, my plan will (generally) pay for 50% of the bill (at the negotiated rate) and I pay the balance. It it capped at $2,500 per year.
From what I've read so far, the new dental plan does not cover anything if you don't use one of their providers. So, it will essentially be useless to me because the participating providers in this area all seem to be fly-by-night operations (located in strip malls) with doctors whose names I can not pronounce. And the couple in the immediate area got bad reviews too