Carole my book of Civil War music, "Singing Soldiers, A History Of The Civil War In Song" lists "Weeping, Sad And Lonely" or "When This Cruel War Is Over" as the same song. The notes on it say '"It is reported that the sale of this song approached a million copies, and that it was very popular with both the North and the South. So depressing were the sentiments of "Weeping, Sad And Lonely" that officers forbade its singing. The Southern soldiers substituted "grey" for "blue"'
I would not take a LOT of Burn's interpretations as gospel. He took more than a few liberties with various things on that PBS program.