The difference, by the numbers
Both have aged rum and lime juice. Only the Mai Tai has Jamaican rum, orange curaçao, orgeat and simple syrup; only the Three Dots and a Dash has rhum agricole, falernum, allspice dram, orange juice, honey syrup and Angostura bitters. In parts, a Mai Tai is 2 rum, 2 rum, 1 Curaçao, 1 orgeat, 2 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 Mai Tai is 47% spirit, 23% citrus, 18% syrup and sugar and 12% liqueur; 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 14.5%) and holds more alcohol (1.79 standard drinks against 1.69). Calories are close (about 242 and 235), but the Mai Tai has more sugar (17.2 g against 14.5 g). The Mai Tai is shaken and served in a rocks glass; the Three Dots and a Dash is blended and served in a goblet.