The advantage of studying a language synchronically is that it allows the linguist to concentrate in a specific setting in which the language will develop and show patterns of growth, trends, and even extinction, rather than comparing the language with a myriad of other times and settings.
When you study synchronically, you are socially "in synch" with the time and place of where the language is being used. You can explore it AS IT is happening, and you can find the trends that occur as a result of it.
A contrasting study would be the diachronically study of linguistics, in which you would have to compare and contrast the development of a language, trend, or dialect back against another time and place, set up timelines, develop historical studies, and enter way deeper into the semantics of it all.
Therefore, the advantage of sychnonic linguistics is precisely that it is a study that can be conducted in the now, or at any point in its past, but does not have to be correlated to any other linguistic occurrence.