Question
what are the advantages of synthetic fertilizers?

Answer
Compared to organic fertilizers, synthetic fertilizers typically have the advantage that they are quick release (useful for cool sesaon crops and in cold weather). Synthetic fertilizers are also often more compact (concentrated) which means less weight to carry and distribute. Since they are pre-formulated in factories, they typically come mixed in the ideal ratios needed for various purposes, such as vegetables, roses, lawns, etc. Organic fertilizers are often "take as they come" and must be combined one or more products to get the ratios right.

Finally, synthetic fertilizers can (but it is not always so) be cheaper than organic fertilizers.