Cynically, social democrat is somebody who doesn't want to spook the centre, a democratic socialist is somebody who wants to spook the centre just a little.
The distinction is fuzzy mostly because the two terms used to be synonyms; the term "social democrat" is I understand more natural to German-speakers than "democratic socialist", and because the SPD were the dominant force in pre-1914 socialism, it became the more common term. When the SPD fractured after WWI, "social democracy" became associated with the party rather than the idea (similar to "communism"), so those to the left of the SPD began identifying themselves as "democratic socialists". The latter term eventually gained additional connotations, "democratic" affirming a commitment to democratic government and thus opposition to Stalinism, and "socialist" affirming commitment to socialist principles and thus opposition to settling for a more equitable capitalism. These associates are retained to some degree, although they're not as strong in distinguishing self-described "democratic socialists" from "social democrats", because nobody cares about Stalin except libertarians and Stalinists, and nobody cares about either libertarians or Stalinists, and because even "democratic socialists" aren't usually committed to socialism any more, just a sort of democratic and egalitarian capitalism.
Today, the distinction is basically about legitimacy. "Social democrats" identify themselves with a tradition that has historically participated in and even lead governments, so it represents a claim to legitimacy as a Party of Government. "Democratic socialists" identify themselves with a tradition that has historically opposed the powers that be, so it represents a claim to legitimacy as a Party of the Masses. Both naturally claim to represent "the People", but the former imagine the People in traditional republican terms, the people assembled in parliament, while the latter imagine the People in classically radical terms, the people assembled in the street. The actual political differences are in some way less important; there's more variation in outlook within each umbrella than between them, and the left-wing of the social democrats parties are distinguished from the centre-wing of the democratic socialists more by their commitment to a major party than by any fundamental political difference.