If you're ever asked what gas either of these terms refers to, the answer required is invariably "methane". But strictly speaking, neither is correct.
According to Wikipedia, marsh gas is "a mixture of methane, hydrogen sulfide, carbon dioxide, and trace phosphine produced naturally within some geographical marshes, swamps, and bogs"; while firedamp "is the name given to a number of flammable gases, especially coalbed methane."
In respect of firedamp, Wikipedia adds: "Damps is the collective name given to all gases (other than air) found in coal mines in Great Britain. [Other] damps include blackdamp (nonbreathable mixture of carbon dioxide, water vapour and other gases); whitedamp (carbon monoxide and other gases produced by combustion); poisonous, explosive stinkdamp (hydrogen sulfide), with its characteristic rotten–egg odour; and the insidiously lethal afterdamp (carbon monoxide and other gases) produced following explosions of firedamp or coal dust.
© Haydn Thompson 2021