Dental Insurance Billing and Braces Cost?

I'm hoping someone can help me with my dilemma. My daughter had braces and when they installed them, they told me the total cost would be $5,800 - my portion would be $2,500 and our dual dental insurance (Delta Dental PPO) would cover the rest. I paid it up front. She had them removed recently, and the dentist (in-network provider) is asking that I pay an additional $600 that my insurance is refusing to pay. 

I called Delta Dental and they said that their "contract amount" for braces with our dentist is $3,700, and they already paid them $3,200. They said I should only be responsible to pay the difference ($500) between what Delta paid and the contract amount, and that I over-paid my portion by $2,000!  

My dentist is saying this is not true and they are willing to waive the additional $600 they are billing me for right now. But now I feel like I've been gouged and want a reimbursement, if I in fact, overpaid.  

Anyone know how this billing thing works?  Delta and my dentist are telling me two different things and not sure who is correct.  Thanks much!