Agreed with all the points above about the differences between JSPs and Servlets, but here are a couple additional considerations. You write:
I have an application that sends the
  customer to another site to handle the
  payments. The other site, outside of
  the customer, calls a page on our
  server to let us know what the status
  is of the payment. The called page
  checks the parameters that are given
  by the payment application and checks
  to see whether the transaction is
  known to us. It then updates the
  database to reflect the status. This
  is all done without any interaction
  with the customer.
Your application is consuming the payment service of another application. Your solution is fragile because if the payment service in the other application changes, that breaks your JSP page. Or if you want to change your application's payment policies, then your page will have to change. The short answer is that your application should be consuming the application's payment service via a web service. Neither a servlet nor a JSP page is appropriate place to put your consumption logic. 
Second, along those lines, most usages of servlets/JSP pages in the last few years have been put inside the context of a framework like Spring or Struts. I would recommend Spring, as it offers you the full stack of what you need from the server pages to web service gateway logic to DAOs. If you want to understand the nuts and bolts of Spring, I would recommend Spring in Action. If you need to understand better how to tier an enterprise architecture written in a language like Java (or C#), I would recommend Fowler's Patterns of Enterprise Application Architecture.