"He cannot work because of the snow."
Translation:Il ne peut pas travailler à cause de la neige.
Is the only difference between "à cause de" and "parce que" their grammar (that one is followed by a clause and one by a noun), or do they have different colloquial usage/meaning?
No, you're right, the only difference is what they are followed by. They both just mean "because".
Is it okay to use "car"? doulingo marked it wrong: "il ne peut pas travailler car le neige"
There are three forms of "because" in french or rather 3 ways of thinking about it; the first one is Parce que, this is the most common and we use it when the phrase is followed by a sentence. Example: J'aime le film parce que j'aime les films d'horreur. The other two are used when because is followed by a noun. "grâce à" is traslated as "thanks to" and the other is negative "à cause de". In this sentence you can see that grâce is followed by au not à. This is another grammer rule, where it would read "grâce à la rébus" but à + la is written as au. Hope this all makes sense :)