To put it simply: socialism is a specific political theory and social welfare is government assistance for basic needs (what basic means can vary from country to country, of course. There isn't a country in Europe these days that is socialist. Now, socialism as a movement and as a political theory had influence on the European welfare states, but that doesn't make them socialism.

The misconception in America stems probably from people like Bernie Sanders that call themselves socialists and praise the social welfare systems of Europe, somewhere this merging in the minds of many Americans as synonyms instead of two distinct things.