The difference, by the numbers
Both have aged rum and lime juice. Only the Banana Daiquiri has banana liqueur, simple syrup and banana; only the Three Dots and a Dash has rhum agricole, falernum, allspice dram, orange juice, honey syrup and Angostura bitters. In parts, a Banana Daiquiri is 4½ rum, 3 banana liqueur, 1 lime and ½ syrup; a Three Dots and a Dash is 3 rum, 1 rum, ½ falernum, ½ allspice, 1 lime, 1 orange and 1 honey. By volume, the Banana Daiquiri is 50% spirit, 33% liqueur, 11% citrus and 6% syrup and sugar; the Three Dots and a Dash is 50% spirit, 13% liqueur, 13% syrup and sugar, 12% citrus and 12% mixer and juice. That makes them close cousins on the ratio map.
The Three Dots and a Dash is the stronger drink in the glass (16.3% ABV against 13.3%) and holds more alcohol (1.79 standard drinks against 1.44). The Three Dots and a Dash has more calories (about 235 against 224), but the Banana Daiquiri has more sugar (17.4 g against 14.5 g). Both are blended; the Banana Daiquiri is served in a coupe, the Three Dots and a Dash in a goblet.