After a timetable-based or headway-based assignment, all departure times are at midnight respectively 00:00:00.
The term connection indicates a time-based information as to say the path has a precise departure time. Routes don't have precise departure times (always 00:00) and they combine all connections with the same course.
The parameter setting for 'Save assignment results' is set to 'As routes'. This saves memory. To get departure times choose 'As connections':
Calculate -> General procedure settings -> PuT settings -> Assignment -> Save assignment results -> Save paths 'as connections'.
In case of a headway-based assignment there are no departure times at all. A headway-based assignment only returns routes as assignment results.