'Movie snow' is made of several different things, each having a different weight & different purpose.
If they'd really wanted it to fall straight downwards, they'd have used something heavier.
I've seen sets dressed for snow and worked on scenes using it, but I don't know the full technical details, so I'm going to have to skirt the 'hard data' a bit.
The basic types I've seen are made of paper [wet or dry], foam, cellulose and formaldehyde.
The very light 'snow' is formaldehyde, burned as candles - that will definitely give the look in the Gladiator clip.
Foam, as far as I know, is used for heavier snowfall.
Paper and cellulose are also used to make snow-fall, but I've never seen those in real life, only on other people's footage.
I've seen it laid down on the ground as a blanket effect prior to the shoot, but not used as fall.
Paper is extremely good for ground-coverage. If you wet it slightly it even holds footprints that look and feel entirely convincing even when they're your own feet making them and it's 30°C in the shade.
Have a look at Snow Business, a UK company, for the myriad ways they have of trying to convince you it's actually snowing.
...and I only just realised, that's the company who did the snow for Gladiator!
See the page on Snow Sticks
After comments - I feel it doesn't change the premise of the question whether it's light snow or actually ash in the first-mentioned Gladiator clip. There are plenty of occasions when show flurries in all directions, unless the weather is completely calm and still.
Different artificial snow weights and/or large fans out of shot can change the appearance on-screen to be whatever the director requires.