What is the difference between the amount in payment received for tuition and the amount billed ?

The 1098-T which colleges issue to students will have a figure in either Box 1 (payments received) or Box 2 (amounts billed). The difference is that Box 2 amounts are not proof of payment in the IRS' eyes.  Box 2 amounts, while billed, may or may not have been paid.

When you use the Box 2 amount to claim an education credit, the IRS will sometimes (more & more these days) send you a letter asking for actual proof of payment.  Then you'll have to respond and show them some documentation proving payment, such as a statement from the school showing those amounts posted, a credit card statement showing the amount & who it was paid to, or a bank statement/cancelled check.  In those cases, the IRS will check your documentation and, if all is in order, approve the education credit.  Basically, it's a hoop you have to jump thru on occasion to get the credit.