2^a-3^b >> (2+\epsilon)^b
Note that Ellison pushed a bit with 2^a-3^b > 2.56^b and using these tricks you can push as far as you want (e.g. with b>2000 we have 3^b-2.99^b > |2^a − 3^b| > 2.99^b, the left inequality relying on a=\lceil b\log23\rceil). These are not helping much.
Rhin pushed (the log version) to a^{-7.616} for some a>a_0, and some are trying to push it to a^{-2} or b^{-2} to Roth’s levels. They would be far tighter, but this may be not enough either…