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. Timetable-based assignment: 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'. Headway-based assignment: In case of a headway-based assignment there are no departure times at all. A headway-based assignment only returns routes as assignment results.