As a ex-bartender, did this while travelling through Europe as a young man, there is both a gin and vodka martini. The difference is the added flavours. That difference is the preference of Juniper berries in your alcohol. When I first learned to bartend the old guy had the Savoy Cocktail book where I was trained. I tried different recipes and I still prefer the 1920s formula of 3:1 Vodka to Vermouth. Shake the darn thing till your elbow is sore and the ice cubes have reduced to ice crystals. Three olives and you have a delicious, clean ice cold drink thats great for the cocktail crowd. To impress the ladies I will sometimes take over the bar and mix the Martinis, Tom Collins, Slings and all the old favorites.