A variation on the meaning posted by Fiolar is given in "A Dictionary of Buckish Slang, University Wit and Pickpocket Eloquence," by Frances Grose, 1811 edition.
Smart money- money allowed to soldiers or sailors for the loss of a limb or other hurt received in the service (better known as smart ticket).

Another is given in the Military Dictionary, James, 1802, quoted in the OED:
"Smarts: The different sums which are received by recruiting parties under the heading of smart money are frequently so called."

The definition posted by Fiolar and the meaning in the song is also in the OED, from Jamieson's Scottish Dictionary, Supp., 1887: "He listed... but paid the smarts and wan hame."