Hi @Muskox,
I recently ran into the exact same issue with Shannon diversity estimates with 2 different programs (except it was with another program, not phyloseq)
uses scikit-bio for calculating these diversity metrics — and it looks like scikit-bio Shannon uses log base 2 not log base e.
Other than that, @colinbrislawn's suggestions re: how phyloseq processes feature tables upstream could add to the dissimilarity in results.
I hope that helps!