Alkylate, a major component of modern gasoline would be considered "synthetic" by the same definition as synthetic oil. The main advantage is higher antiknock index (octane) rating without lead. This allows for higher compression ratios, resulting in more power and mpg's. Another advantage is high purity, preventing the gum and varnish deposits that used to be common in fuel systems.
Ethanol is produced by simple distillation, hardly "synthetic", but Methanol is made from natural gas in a way that could be called synthetic. Fuels made from coal or natural gas by the Fischer Tropsch process are synthetic, but you get diesel and jet, not gasoline, without a lot of further processing.