Saying that a seed grows from the Earth is correct, and so is saying that a seed grows from the soil.
In the context you stated, both words refer to the ground. Saying that the seed grows from the ground/soil is more technically correct with regards to science, but some English speakers call the "ground" the "Earth," either in a joking way, or just as a common expression.
The main difference between the actual definition of the words is that "Earth" refers to the planet we live on, while "soil" is another word for dirt where you can grow plants/crops.