Why do both Ramsey and John Snow have that last name?

Highborn bastards take on the regional surname of their acknowledged parent. Ramsey and Jon both have the same surname because they are acknowledged bastards of highborn noblemen of the North. “Snow” is the bastard name for Northern bastards. read more

The name varies by region: in the North, for instance, it's Snow, but it's Pyke in the Iron Islands, Storm in the The Stormlands, Hill in the Westerlands, etc. So although Jon and Ramsay may share a surname, they're most definitely not related. Jon, as we all know, was the bastard son of Ned Stark, while Ramsay was born to Lord Roose Bolton. read more