Shorter/narrow pulse width results in finer resolution but shorter range. A wider pulse has lower resolution but longer range. For experimental weather Radar different pulse widths uses are many. Could be one pulse width is for reflectivity and the other pulse width is for velocity. I am not an expert but have worked with some weather radar data sets in different forms before... So this is just a guess. Encoding may also be a factor if used.
Solving range and velocity ambiguities is often accomplished by comparing target data acquired over two, or more PRFs (not pulse widths). A burst of pulses at a fixed PRF may be transmitted and processed within one coherent processing interval (CPI), and then the radar shifts to another CPI on a new PRF; the process may use up between 5 to 9 PRFs in a medium PRF mode (8 PRFs is commonplace), or fewer in low and high PRF pulse Doppler modes. Changing the pulse width in sympathy with the change in PRF enables the radar to operate on the same duty ratio and this has several advantages for the radar. Firstly, a constant duty ratio maintains a constant average power and so the same energy per CPI (assuming equal number of pulses per CPI) and this, in turn, yields a constant detection performance over the various CPIs. Secondly, it may be advantageous to run a high power transmitter at a constant load and so maintain the same average power (if the pulse width didn't change with PRF, the load would vary between CPIs). This is more of a practical engineering point in designing transmitter power supplies. This is perhaps most likely to be the reason why a radar uses several pulse widths but there are other reasons. Changing pulse width to maintain a constant duty ratio brings about a few complications so not all radars will do this.
One of the other reasons for changing the pulse width may be that a shorter pulse is used at shorter ranges (and a longer pulse is used at longer ranges). The shorter pulse has less energy and so a reduced detection performance but this is acceptable at short ranges. Also, a narrower pulse width has a shorter blind range (and close-in eclipsing losses) and this may be important in order to maintain target visibility and minimise the effects of partial eclipsing at very short ranges.
There could be other reasons for using differing pulse widths but the two described above are perhaps the most likely. I could say more if you could tell me more about the radar and its waveforms.
This might help: https://shop.theiet.org/pulse-doppler-radar
I agree with Clive Alabaster , Weixian TAN , Reuben Reyes , and Ilyas Evgenyevich Varshavskiy
The shorter pulse has less energy and so a reduced SNR at detection and estimation signals parameters of long-range targets. But at a short distance need at shorter pulses, because SNR is enough. Another context, which was considered before, take to account as well.