The Flickr example you link to is of a single stationary snowflake, standing on two points so that it is crisply illuminated from behind. If you can arrange a snowflake like that all you need is a macro lens, and most phone lenses are capable of decent macro shots like that.
The problem with snow is that it usually falls as clumps of snowflakes. Go outside and shoot a "snowflake" that has hit the ground and you will see that.
You can capture crisp falling snow if your shutter speed is fast enough relative to the wind and fall rate — probably high hundredths or thousandths of a second. But if you want a macro shot of falling snow you need a flake/clump to be both close to your lens and inside your focal field when the shutter trips. In a heavy storm you can probably pull that off if you take a lot of shots.