set theory - Connection between the axiom of universes and Tarski's axiom

Over ZFC, the theories are equivalent. Both Grothendieck's Universe Axiom and Tarski's Axiom are equivalent to the assertion that there are unboundedly many inaccessible cardinals.

Grothendieck universes are exactly the sets $H_kappa$ for an inaccessible cardinal $kappa$, consisting of all sets hereditarily of size less than $kappa$. They can also be described as $V_kappa$ for inaccessible $kappa$, using the cumulative Levy hierarchy. The Tarski universes, in contrast, needn't be transitive, and so the notions are not equivalent.

The issue is that over ZF, they lose their equivalence. Solovay explained on the FOM list that this is due to the way that cardinalities are treated in TG, making ZF+TG imply AC, whereas ZF+GU does not imply AC. You can read Solovay's interesting post here.