The difference, by the numbers
Both have vodka and coffee liqueur. Only the Espresso Martini has simple syrup and coffee; only the White Russian has cream. In parts, an Espresso Martini is 1¾ vodka, 1 coffee liqueur, ¼ syrup and 1 coffee; a White Russian is 2 vodka, 1 coffee liqueur and 1 cream. By volume, the Espresso Martini is 42% spirit, 25% mixer and juice, 25% liqueur and 8% syrup and sugar; the White Russian is 50% spirit, 25% mixer and juice and 25% liqueur. That makes them close cousins on the ratio map.
The White Russian is the stronger drink in the glass (21.1% ABV against 13.6%) and holds more alcohol (1.69 standard drinks against 1.47). The White Russian has more calories (about 323 against 223), but the Espresso Martini has more sugar (19.6 g against 14.4 g). The Espresso Martini is shaken and served in a martini glass; the White Russian is built and served in a rocks glass.