What people are hoping for today is to get CO₂ from the atmosphere and hydrogen from H₂O with electricity from solar, wind or nuclear. (Though w/ nuclear you might use thermochemical hydrogen if you can get the temperature up)
The best use case for that I've seen is that the US Navy would like to synthesize jet fuel on aircraft carriers so they don't have to slow down to take on fuel from a tanker -- delivering fuel to an aircraft carrier has to cost more than it costs at your local gas station.
Also I think there is more interest in synthetic fuel for airplanes than cars. That Mazda could almost as easily have been run on ethanol, methanol or maybe even butanol. Even methane isn't that hard. Diesel engines can be run on Dimethyl Ether. Single-entity fuels are more efficient to make than synthetic fuels based on long-chain hydrocarbons.
Aviation, on the other hand, is a place where innovation goes to die, where airlines struggle to replace obsolete electronics, where the #1 and #2 aircraft in the world were designed in 1967 and 1982, etc. (... it wouldn't even be legal to sell a car based on a 1982 design!)
