Diachronic linguistics is the same thing as historical linguistics. Diachronic linguistics is the study of the changes in language over time. Synchronic linguistics is the study of the linguistic elements and usage of a language at a particular moment. Diachronic analysis can be the general evolution of all languages or the evolution of a particular language or dialect.
Think of a sychronic analysis as a single frame in a roll of film. The diachronic analysis would be the study of all the frames.
In Course in General Linguistics, Ferdinand de Saussure discusses how an individual can contribute changes in a language but without the acknowledgment of this change by others in his/her social community, no change will have lasting value. In other words, he says "by himself the individual is incapable of fixing a single value."
Saussure favored the synchronic approach. Each approach has its benefits. The diachronic approach might focus more on the interplay of historical events and language changes. The synchronic approach of English in America today would look at things like texting, e-mail, and all modes of expressing language in order to see how those manifestations of language reveal the structure of a language.
The diachronic approach is historical (compare with Continental Philosophy) and the synchronic approach deals more with the system/structure of language (compare with Analytic Philosophy). If you were doing a diachronic analysis of American English in El Paso, you would want a length of time to make an historical analysis: from 10-100 years or more. If you were to do a synchronic analysis, you might pick one year or even less time. The point is that a diachronic analysis is not just doing more research; it is doing something different.
Saussure believed that a good synchronic approach made a diachronic approach unnecessary. He used the analogy of chess. As a chess game progresses, the value of a piece changes (diachronic). But using a synchronic approach, one could determine how the pieces interact at a given time, which is an expression of the rules of chess (grammar or rules of language). Recent theorists acknowledge the benefits of this approach as well as the historical analysis of the diachronic approach. In other words, a diachronic analysis shows the evolution of the game, how the pieces interact and how their values change. The synchronic analysis focues more directly on how the pieces interact to reveal the rules of the game.