The main musical difference between a short drag and long haul chantey is that the choruses to a short drag are single (since there is one haul per verse) and a long haul has two choruses per verse. For example, Sally Racket, a short haul chantey, has the one chorus "Haul 'er Away!" after each verse, representing the one pull on the line done between verses. The verses themselves are very short one-liners - "Oh, Little Sally Racket (Haul er Away!)," etc. A long haul chantey, such as the afortementioned John Kanaka, has two pulls on the line, on the words "John" and "Tu."
Capstan chanteys tend to (but not all) have a "grand chorus" which is much longer than hauling chantey choruses. For example, after a couple of short call and response verses/choruses, Hieland Laddie's grand chorus is "Way, hey, and away we go, bonny laddie, hieland laddie. Way, hey and away we go, my bonnie hieland laddie."
As you know, many chanteys were interchangeable as long as they fit the job at hand, but that's what musically separated many.
Hope that helps, and feel free to PM me if you'd like to go into it more.