The Chi Square test (Goodness-of-Fit test) will be very good at comparing the tails of two distributions since it is structured to compare two distributions by buckets of values (graphically represented by a histogram). And, the tails will consist in the far most buckets.
Even though this test focuses on the whole distribution, not just the tail you can readily observe how much of the Chi Square value or divergence is derived by the difference in the tails's fatness.
Watch that the derived histogram may actually give you visually much more information regarding the respective fatness of the tails than any test related statistical significance. It is one thing to state that tails fatness are statistically different. It is another to visually observe it. They say a picture is worth a thousand words. Sometimes it is also worth a thousand numbers (it makes sense given that graphs encapsulate all the numbers).