The difference, by the numbers
They have no ingredient in common: the El Presidente has white rum, blanc vermouth, orange curaçao, grenadine and Angostura bitters, the Martinez gin, sweet vermouth, Maraschino liqueur and orange bitters. In parts, an El Presidente is 1 rum and 1 blanc vermouth; a Martinez is 1 gin and 1 vermouth. By volume, the El Presidente is 46% spirit, 46% fortified wine and bitter, 5% liqueur and 3% syrup and sugar; the Martinez is 48% spirit, 47% fortified wine and bitter and 5% liqueur. Different bottles, similar balance: that makes them close cousins on the ratio map.
They are about equally strong in the glass (19.5% and 19.7% ABV), and they hold about the same alcohol (1.55 standard drinks each). Calories and sugar are close: about 188 and 184 calories, 8.4 g and 8.0 g of sugar. Both are stirred; the El Presidente is served in a martini glass, the Martinez in a coupe.