A cup is rounded and has a handle, a can is straight-sided and doesn't have a handle. I don't know what is significant about the difference as far as the song is concerned, except that a cup was probably higher class. That would be in line with the song's running joke, which is that gentleman Martin is getting drunk with his servant. But see Steve Howlett's post of 16 Jul 17 above for a more imaginative interpretation.

Cans are occasionally mentioned in songs involving drink; e.g. The Hsarvest Horn

The master brings the can, he's a jolly-hearted man,/ "Come my lads and take a pull of the best" etc .....

The earliest record of the song was when it was registered by a Thomas Orwyn in 1588, so it has some history.