The singer meets and courts a girl with fine hair tied up in a (black/blue) velvet band. As they are out (walking) one night, she steals a gentleman's (watch). The crime is discovered; she plants the evidence on the singer; he is convicted and punished
Roud splits this into two songs, based perhaps on whether the band is black (#2146) or blue (#3764). It may well be that the "blue velvet band" versions are a rewrite. Certainly the version produced by Spaeth is the sort of thing you'd expect when someone "improves" a traditional piece: The stanza form is different, and it's full of cutesy forms.
But it's the same story, and the "blue" form is less popular, so I'm content to lump them while considering the blue velvet band secondary and the result of redaction.
It should be noted that the fullest versions of the "Blue" version, such as Spaeth's, are extremely full, with (in effect) two plots: First the wild meeting which results in the young man being convicted and punished, and then a final scene in which the young man misses the girl and goes to find her, only to find her dead. There is another "Blue" version (in the Index as "Blue Velvet Band (II)" ) in which the middle part, about the prison, has broken off. Genetically, it's still the same song, and perhaps should file here -- but the parts have separated so far that it seemed better to split them.
In any case, there are so many black and blue velvet bands floating around the tradition that you probably should check all songs which use these titles.
Inceidentally, it seems pretty certain that the song was well-known in the ninetheenth century; according to Spaeth's _A History of Popular Music in America_, p. 608, there was a popular piece of 1894 entitled "Her Eyes Don't Shine Like Diamonds" by Dave Marion. - RBW