I had a bad experience with UA CS. I wrote an angry letter about the shabby treatment I received from their customer service. I included one small reference to their poor quality of English in my rather length letter--it was extremely hard for me to understand what they were saying. Their response was only directed towards my comment about their agents' English skills, and ignored all the other bigger issues. I guess they went on the defensive regarding that issue.
AA CS has been consistently good and helpful. I've never had a problem communicating with their aagents. They also seem to be friendlier and have more personality than those at UA. I'm usually never on hold for more than 5 minutes...unlike UA.