This week's book giveaway is in the OCAJP 8 forum. We're giving away four copies of OCA Java SE 8 Programmer I Study Guide and have Edward Finegan & Robert Liguori on-line! See this thread for details.
There are a lot of web service stacks and I suspect that there is not one that is best in all situations.
A suggestion is to give yourself a small example assignment of some sort.
First you solve the assignment with Axis 2, then you solve it with Apache CXF and finally you come back here and write about your experiences!
This would give you insight in how these two web service stacks work in the area of your interest, instead of just trusting me, or someone else which experiences may not match your area of use.
By the way, Apache CXF also is a mature web service stack originating in 2008 from the merging of the Celtix and XFire web service stacks.