While these two terms are very similar looking and indeed can both be traced back to the early socialist movement in Europe and America, today they represent very different methods for approaching the issue of socialism. As such I think they should both be defined and then compared in order to clarify the differences and similarities.
This is more difficult of the two the nail down in terms of definition, so let me give it my best shot, I will focus on the definition that most distinguishes it from social democracy. Among those definitions of democratic socialism which sharply distinguish it from social democracy, Peter Hain, for example, classes democratic socialism, along with libertarian socialism, as a form of anti-authoritarian “socialism from below” (using the term popularised by Hal Draper), in contrast to Stalinism and social democracy, variants of authoritarian state socialism. For him, this democratic/authoritarian divide is more important than the revolutionary/reformist divide. In this definition, it is the active participation of the population as a whole, and workers in particular, in the management of economy that characterises democratic socialism, while nationalisation and economic planning (whether controlled by an elected government or not) are characteristic of state socialism. A similar, but more complex, argument is made by Nicos Poulantzas. It should also be noted that under this definition, democratic socialists tend to support revolutionary means and methods as opposed to reformist/evolutionary ones.
Social democracy is a political ideology that emerged in the late 19th century out of the socialist movement. Modern social democracy is unlike socialism in the strict sense which aims to end the predominance of the capitalist system, or in the Marxist sense which aims to to replace it entirely; instead, social democrats aim to reform capitalism democratically through state regulation and the creation of state sponsored programs and organizations which work to ameliorate or remove perceived injustices inflicted by the capitalist market system. The term itself is also used to refer to the particular kind of society that social democrats advocate.
Comparing the two:
One way to delineate between social democratic parties and movements and democratic socialist ones is to think of social democracy as moving left from capitalism and democratic socialism as moving right from Marxism: in other words, a mainstream leftist party in a state with a market economy and a mostly middle class voting base might be described as a social democratic party, while a party with a more radical agenda and an intellectual or working class voting base that has a history of involvement with further left movements might be described as a democratic socialist party. However, this is not always the case. The British Labour Party charter identifies the party as a “democratic socialist party,” even though the current and former leader, Gordon Brown and Tony Blair, self-identify as social democrats.
The terms democratic socialism and social democracy have often been used interchangeably, and many have considered them synonymous until recently. Now the term social democracy refers to an ideology that is more centrist and supports a broadly capitalist system, with some social reforms (such as the welfare state), intended to make it more equitable and humane. Democratic socialism implies an ideology that is more left wing and supportive of a fully socialist system, established either by gradually reforming capitalism from within, or by some form of revolutionary transformation.
The tension between the revolutionary and evolutionary tendencies of democratic socialism can be seen in the Socialist Party USA, which has members who advocate both types of positions (although the party statement of principles includes the word “revolution”). Revolutionary democratic socialists accuse those who favor evolution of supporting socialism from above, which does not abolish the capitalist system. Revolutionary democratic socialists believe that the political structures within existing capitalist societies serve as an impediment to full democracy, which they believe can only be achieved by establishing a new political structure built from the bottom up. Evolutionary democratic socialists accuse supporters of revolution of being impractical.
Evolutionary (reformist) democratic socialists and social democrats both typically advocate at least a welfare state, although some social democrats, being influenced by the Third Way, would be willing to consider other means of delivering a social safety net for the poorest in society. Revolutionary democratic socialists support a welfare state not as a means of achieving socialism, but as a temporary method of relief, and as a means of mobilizing the populace towards revolutionary ideals. Democratic socialists usually support re-distribution of wealth and power, social ownership of major industries, and a planned economy. Social democrats have largely abandoned these concepts. Many democratic socialists retain a Marxist analysis, while social democrats might entirely reject Marxism.