How does contrail formation differ from turbofan to turbojet?
It appears that the higher bypass engines available today increase the contrail occurrence. From Influence of propulsion efficiency on contrail formation by Ulrich Schumann:

The  thermodynamic  analysis,  which  is  the  result  of first-principle  arguments,  implies  that  aircraft  and  engines, performing with a higher overall propulsion efficiency release a smaller fraction of the combustion heat during cruise  into  the  exhaust  plume, and  hence  cause plume conditions which during mixing reach higher relative  humidity  for  the  same  ambient  temperature  and hence form contrails also at higher ambient temperatures.
Hence aircraft will form contrails more frequently when
using more fuel efficient engines.

TO test the theory, a flight test was conducted between two aircraft with low and high bypass engines:

For a direct test of the theory, a formation flight of two different large  jet aircraft was arranged, wing by wing, during an ascent and a descent f the aircraft. Contrail formation and ambient conditions were observed simultaneously from a research aircraft.
The two contrail forming aircraft were
(i) a Boeing B707 equipped with four jet engines of type JT3D-3B with bypass-ratio of 1.4 and
(ii)  an  Airbus  A340-300 with  four  jet  engines  of  type
CFM56-5C4  with  bypass-ratio  of  6.8.

The results indicated that contrails were formed in more instances in case of the A340 (with higher bypass engines) compared to the B707.

Photo of an A340 with contrail and B707 without one. Image from Influence of propulsion efficiency on contrail formation by Ulrich Schumann

Ambient  conditions were measured and the contrail formation was observed from a research aircraft flying less than 1 km behind the two contrail forming aircraft. As documented in several photos, an altitude range exists in which the A340 causes  contrails while  the  B707  causes  none.

Though the sample size is pretty small, it does appear that the high bypass engines used in modern commercial aircraft are more prone to contrail formation compared to the older turbojets.