Adding a payment is acknowledging that a payment has been received. When adding a payment, the funds will automatically be deducted from the outstanding totals. Applying a payment allows you to keep track of payments against certain invoices. Applying a payment has no effect over the totals of a client, it mainly just changes an invoice status from saying Unpaid to Paid. There are some benefits to applying payments as far as Reporting goes. Invoice AR Aging, Payments Applied, Collections, and Invoice report all require payments to be applied to Invoices in order to show proper totals on these reports. There are other reports like Account Activity, Account Balance, Payments Received, etc., that do not require you to apply payments to invoices in order to run properly.
For instructions on how to apply payments click here.