What would I be looking for on the oscilloscope?
How sharp is the signal, how steep the edges, what's the signal level, does it have spikes and the like. Is the timing correct or are there stretches that shouldn't be, etc. The hardware serial tries to adapt if the signal doesn't have the correct speed (within some limits) while the SoftwareSerial just reads in defined time periods.
Are the characters read by SoftwareSerial correct (the ones sent)? Or do you just look if something is transmitted?