It's snowing heavily today.
'snow shower' means: It's snowing lightly.
'rain shower' or 'shower' means: It's raining lightly.
'There is a snow flurry.'
or
'There are snow flurries.'
or
'It's flurrying.'
means: It's snowing heavily.
'flurry' can also be used to mean: a lot of other things
for example, in boxing: a flurry of punches or a flurry of blows
more examples: a flurry of attacks, a flurry of complaints, a flurry of activity
