JTAPI stands for Java Telephony API. It was developed by Java for Computer Telephony Applications or CTIs. It provides a set of packages which provides basic guidelines for placing, answering and dropping a call. JTAPI is similar to TAPI (Telephony API) which was developed by Microsoft and Intel.
AXL stands for Administrative XML. AXL is used for pushing config on another system.
When integrating another system with CUCM (for example UCCX), JTAPI is used for CTI (placing call, ending call), and AXL is used to push the config onto CUCM.