Recurring invoices allow you to automatically send a duplicate invoice at a certain point in time [every (#) of weeks/months/years] that ends after so many occurrences. You will select the email template that is used (same for all invoices that are sent in this recurrence), and create the invoice (Same invoice sent each time for this recurrence). This works well for users who want to bill for monthly services, that never change.
Payment schedules are a way to create custom due dates for different portions of the total invoice. So for example: an invoice of $10,000 can have $2,000 due upon Receipt, and 50% due 2 months prior, and remaining balance due 1 week prior to the project. These payment schedules can be either equal or custom.