Synchronic linguistics asks questions like "what does this word mean NOW?" and "How do people use this structure NOW?" Diachronic linguistics asks questions like "How has this word changed over time?" These days, one of the most important roles of diachronic linguistics is figuring out how different languages are related to each other.

The difference is related with time. While diachronic studies the language in its development, across, through and between time (historical changes, linguistic variations through time, etc.), the main task of synchronic linguistics is not to delve into the language history, but to study the language at a given period of time - it has to do with the relationship between words, vocabulary, pronunciation - how people use the language now, for example, dialects that exist now, usage of words by people with different backgrounds (without comparison). Hope that answers your question :)