There is some truth in that (sort of). There’s an 18th century Welsh proverb ‘Eira mân, eira mawr‘ which loosely translates as ‘little snow, big snow’....
When you have big snowflakes it means conditions are much more marginal and it’s much more likely to turn to rain or melt as there’s probably milder air trying to mix in...Smaller flakes usually means that the cold air is much more entrenched and the snow would last for longer like with the beast from the east in 2018 when we had fine powdery snow but it accumulated significantly