I intend to use 2 channels to collect signals from 2 transducers. One is a transmitter and the other one is a receiver. I want to calculate the time difference between these two channels in order to calculate the signal traveling distance. Because the traveling distance is very short, I would like to know if the synchronization error among channels will affect my result or not.
Any errors will always effect a measurement, but are only significant if they are larger than the maximum error that you can tolerate for your measurement and analysis.
With oscilloscopes, the relative limits of vertical resolution and vertical accuracy are different to the relative limits of horizontal resolution and horizontal accuracy. What I mean by that is that, because of the hardware used in the sampling process, voltage resolution can be resolved to smaller values than that of voltage accuracy, i.e. the error in voltage (due to noise, distortion, etc) is typically larger than the smallest voltage step that you can measure. In contrast, time accuracy can be resolved to smaller values than that of time resolution, i.e. the error in time accuracy (due to jitter, aperture delay, etc) is typically smaller than the smallest time interval that you can measure.
So, to correct for voltage accuracy errors you can resort to techniques such as sacrificing a little of the voltage resolution to improve accuracy (e.g. adjusting step values/sizes to correct errors from calibration). But, the smallest resolution in time by which you will be able to differentiate 2 signals will be one sample interval. As any timing errors, resulting in phase offsets between channels will be much smaller than that, your worst case accuracy will be one sample interval (which can't be improved).
So, regarding the PicoScope you're considering, what you need to decide is 'if the sample interval, for the minimum rate at which you need to acquire the data, will adversely affect your measurement accuracy requirements', and if so would a faster PicoScope meet them?