In brief, aerodynamic 'fog' or visible vapor are caused when the local acceleration of the air around the aircraft drops the static air temperature - if the relative humidity is already high, that drop in SAT can cause moisture in the air to condense forming fog. Aerodynamic fog is visible immediately behind the aero surface that is creating it. Further, although there are exceptions, aerodynamic fog normally dissipates quite quickly - as the air settles down the SAT returns to it's original warmer value and the moisture can 'evaporate' back into the air.
Engine 'fog' - commonly known as contrails - are due to water added to the air from the engine exhaust as a product of combustion - as megan posted the combustion of a ton of jet fuel produces over a ton of water. Engine fog is generally not visible immediately behind the engine since the exhaust is hot - only becoming visible as the exhaust cools to the surrounding temperature. Since engine fog is generated by water being added to the surrounding air, in the right conditions engine fog can persist a long time - sometimes for hours.
Of course engine fog isn't just formed behind aircraft - cars and trucks can generate it (usually when it's really cold). Years ago I was driving across central Utah in the middle of winter when it was extremely cold - around -35 deg. C. At one point when I looked in the mirror, all I could see behind was a white cloud emanating from my car. I was briefly worried that something had just gone seriously wrong with the engine, but a quick check of the gauges showed everything normal - it was just the moisture in the engine exhaust turning to fog in the bitter cold. A few miles further on the ambient conditions changed enough that the fog pretty much disappeared.