Why does silver not react with sulphuric acid but with sulphur in the atmosphere?

For the same reason that carbon will burn in the presence of pure CO but not with pure CO2.

Different molecular properties.

Sulphuric acid is H2SO4 (the sulphur atom being in the centre of the molecule).

It maybe the case that silver is less reactive to the hydrogen or oxygen that is bonded to the sulphur. Everything reacts, just to different degrees depending on the conditions.