Ok, from someone who lives in a snowy place with a toddler-
You need to layer the clothing, I would recommend you get a decent base layer (thermals) and a good fleece, then obviously the snowsuit. Patagonia are fab, though not cheap but definitely good quality.
My daughter lives in this fleece http://www.patagonia.com/web/eu/product/baby-synchilla-fleece-cardigan?p=60088-0-697
and these thermals http://www.patagonia.com/web/eu/product/baby-capilene-baselayer-3-set?p=60051-0-492
Then I would second the recommendation for sallopettes/dungarees and a jacket rather than an all-in-one. We had some columbia ones last year which were good.
You will also need snowboots, we have the kamik snowbugs and they are very good, nice and easy to get on a wriggling toddlers feet http://www.rei.com/product/772623
Then wool socks are a must- never ever cotton in the snow/cold as it gets cold and wet and reduces your body temperature. And either fleece or wool mittens and hat. We find we don't really use a scarf as long as you have a coat that does up to the neck and a hat which come down over the ears and velcors under the chin.
In the Uk I have also found good bits in tk maxx, for adults too.
HTH x