The differences aren’t as striking with regard to pitching evaluations between the two systems. WAR ranked the Chicago White Sox the #2 staff in the league even though they gave up 49 runs above league average. This should have made them a below average pitching staff — not top tier as WAR numbers suggest. The White Sox gave up almost 30 unearned runs above the league average which means they had bad fielding. Perhaps this skewed WAR rankings across the board for the White Sox.
I did not find any large ranking discrepancies in the top 100 ranked WAR and WAA pitchers for CHA. There were only two and they are shown in the tables below.
Chicago White Sox
I suspect the large amount of unearned runs CHA gave up caused an across the board increase in WAR pitching evaluations for all CHA pitchers that incrementally caused the anomaly giving them such a high team pitching rating.