Could someone kindly explain the difference between anticipated payment and advance payment?

Anticipated statuses come based on the no. of steps you want to carry out for a payment. Refer to Form 2 in Maintain payment Method sessions. If you issue a cheque to a supplier and wait for a confirmation from the bank to see if it has been realized or not till that stage it remains as an Anticipated Payment.

Advance is as usual payment without reference to an invoice.

In advance payment, you are directly passing a entry in bank account.
In anticipated payment , you are waiting to clear a check in bank & system.