When comparing acidic strength, compare the stability of conjugate base. One with more stable conjugate base will be a stronger acid.
When $H+$ will leave, a negative charge will be left on the carbon. And remember:
Lone pair and negative charge can't go in resonance due to high repulsion between electrons of both
Stabilization due to resonance is greater than that due to inductive effect
While drawing resonance structures, both of them will have resonance with the +ve charge on the $\beta$-ketone, so the only difference will be due to S and O. Sulphur and oxygen both have lone pair, but it can't participate in resonance.
But S has empty d-orbitals which can do back-bonding, and thus participate in response while oxygen cannot do so. And because resonance trumps over inductive effect, it even if Oxygen is more EN, will not stabilize as much as by Sulphur's back-bonding.
Thus $H_2$ is more acidic.