Apache CXF is a JAX-WS stack, Mule is a lightweight Enterprise Service Bus, they are not comparable. So the question is, do you need an ESB (i.e. do you need something providing a solution for transport, routing, mediation, orchestration, monitoring, etc of messages between applications)?
Given your requirements and given the fact that you're asking what to choose, my suggestion would be to keep things simple and to use Apache CXF. If the need should arise, you will be able to introduce an ESB later.