As far as the contrails are considered, there is little difference between the ones formed by different aircraft. Contrails form when the ambient air (which is usually cooler) mixes with the hot, humid exhaust. This usually happens at high altitudes.
One difference may be that the different particulate matter (eg. soot) emitted by both the engines may be different. These particulates act as condensation nuclei for the water vapor at high altitudes. From FAA's Aviation Emissions, Impacts & Mitigation A Primer:
The soot particles at cruise altitudes interact with other chemicals such as sulfuric acid and nitric acid to form small particles that act as nucleating sites for condensation of water vapor present in the upper atmosphere under certain conditions to form larger particles to form condensation trails or contrails, for short.
While I'd venture that the (older) piston engines emitted more of these particulates, the only study i could find online indicates that the particulate emission is similar in case of gas turbine engines and aviation piston engines using unleaded petrol (while significantly higher in case of older, leaded gasoline, as was used during WWII).
Both images from Microphysical and Chemical Properties of Nanoparticles emitted by Flight Engines by Claus Wahl, Theo Rindlisbacher and Lars Hjelmberg
This would mean that the effect of particulate emissions on contrail formation would've been the same. However, there is another factor- altitude. In general, the (modern) jet engines fly much higher than their piston engines counterparts- the contrails you see in WWII bombers are formed by supercharged engines flying at high altitude to escape from AA defenses. Also, as @ymb already noted, the size/shape and number of exhausts also vary between the piston and gas turbine engines.
Though the mechanism of contrail formation is the same in the piston and gas turbine engines, the difference between them lies in their size/shape per engine and their operating envelope.