Hi!
Jersey is a REST web service stack that aids in implementing RESTful or RESTlike web services.
Mule is an
ESB, which also support routing, transformation etc of messages.
If you are to develop a RESTful web service, then use Jersey. If you are to develop an integration solution that allows multiple systems or components of one single system to communicate with each other, then I would use Mule.
Mule allows you to expose a RESTful web service, but may be overkill if that is all you want to do.
Mule configurations are Spring configurations, so integration with an application that use the Spring framework is very easy.
Best wishes!