What is the difference between a budget and a payment method?

Budgets are not a payment method.  They are a spending limit that applies to all available payment methods.  A site that uses a budget will also need a payment method enabled.  The payment method determines how the user will actually pay for the order.  The budget determines how much the user is able to spend within a given period of time.  At the time of check out, the user will need to select the budget to which the order will apply and therefore deplete the allotted budget by the order amount.