Adam is right. For weather like snow and rain, we don't use an article when we're not talking about any particular snow, just "snow" in a general sense. "Snow fell on the ground. The snow was white." In the second sentence, you have to use "the," because now you're referring to the snow you described in the previous sentence: the snow that fell on the ground. We say "the ground" when referring to the surface of the earth. I think Adam's right that this is pretty much just idiomatic, but I guess in some sense it's true that "the ground" refers to the "particular" ground that is the surface of the earth. If you simply say "ground," it sounds like you mean "some ground," and then the reader is wondering which part of "the ground" you're referring to. We always use "the ground" to refer to the surface of the earth in a general way.

One more thing. Everyone knows where snow comes from, so languages have different rules just for weather phenomena. All languages do. "Spaghetti fell on the ground" is grammatically correct, but nobody will know where the spaghetti came from. Here I say "the spaghetti" because I mean specifically the spaghetti that fell on the ground in that previous sentence. But if I say "snow fell on the ground" everyone knows where the snow came from, so there is no need for further explanation.

It's idiomatic. What that means is there isn't really a rule. You just learn it from examples. If you say "the snow" it sounds like you mean a particular bit of snow. It's like, "that snow right there!" On the other hand, there is only one ground---it's "THE ground". When you talk about snow falling on something, you usually don't mean to specify which snow. It's just some snow. So we might say "some snow", but usually we don't have to say "some". "Some spaghetti" = "spaghetti". "Some chalk" = "chalk". "Some snow" = "snow".

"Snow" is a (mass) noncount noun. We don't use the word "the" in front of a noncount noun UNLESS we are talking about a specific type of that noun. Snow is usually white. The snow in front of my house looks so beautiful (particular snow) "the ground" = the surface of the Earth, or the soil on and under the Earth surface In winter, the ground is usually frozen. After it rains, the ground gets wet and muddy

"the" is a definite article in English. You could say, "The snow covered the ground." It would still be right, depending on the context that surrounds the statement. We would have to read about the snow in a previous sentence. In your example, we can assume that there is probably snow in multiple places and not only covering the ground. We could also assume it might be the first time "snow" has been mentioned to the reader. However, we do know now that it is covering the ground. The ground will have the definitive article "the" because it is the only ground (solid surface of the earth) in the area. To explain this in an example: We can say, "We have snow everywhere in the city" Now we can say after that sentence, "The snow that is covering the ground is grey from the cars driving over it" Hope this helps and makes sense.