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.
