The difference, by the numbers
Both have dry sherry. Only the Bamboo has dry vermouth, orange bitters and Angostura bitters; only the Sherry Cobbler has sugar, orange juice and lemon juice. In parts, a Bamboo is 1 vermouth and 1 dry sherry; a Sherry Cobbler is 14 dry sherry and 1 sugar. By volume, the Bamboo is 100% fortified wine and bitter; the Sherry Cobbler is 93% fortified wine and bitter and 7% syrup and sugar. That makes them close cousins on the ratio map.
The Bamboo is the stronger drink in the glass (13.6% ABV against 11.4%), but being smaller it holds about the same alcohol (0.95 and 0.91 standard drinks). Calories are close (about 100 and 109), but the Sherry Cobbler has more sugar (4.4 g against 1.6 g). The Bamboo is stirred and served in a coupe; the Sherry Cobbler is shaken and served in a julep cup.