The drops are lit by strobed light, and the duration of the strobed light is controlling how well the rain drops are stopped, not the shutter speed. In this case the drops are streaks rather than spots.
I used a hot shoe mounted SB-600 on full power which makes the flash duration 1/900 (approx).
The drops are streaks and lit, which is why they look more like snow. If you used a shorter flash duration the drops would not be as elongated (nor as apparent).
To get an even shorter flash duration to make the rain drops more spot like, you would have to lower the flash power.
At 1/2 power the flash duration would be 1/1600 (approx).
With less power you need to open the lens aperture (1 stop) to have the same exposure of the rain drops, the background would also brighten somewhat with a wider aperture, so to compensate for that you would need to increase the shutter speed slightly to let in less background light. (increasing shutter speed shortens the length of time the shutter is open)
When using flash as the main light source, flash duration is what stops most motion in the scene. Aperture controls the foreground or subject exposure and the shutter speed controls the background exposure and ambient light motion blur.
One last note - any motion blur a strobe lit image has, will precede the direction of motion if the strobed light is triggered at the begining of the exposure (front/first curtain sync), and will trail the direction of motion if the strobed light is triggered at the end of the exposure (rear/second curtain sync).
Most dSLR cameras are set to front/first curtain sync by default.