What is the difference between compost soil and normal soil?

The high content of nutrient rich organic matter tends to be the difference between compost and soil. Compost isn't actually soil. Instead, it represents the aerobic breakdown of compostable materials into the basic elements from which they're made. The most common elements tend to be carbon and nitrogen.

Aerobic means of encouraging the natural decay processes are brought about by careful attention to appropriate levels of air, heat, and moisture. If proper procedure is followed, then compostable materials turn into dark brown, fresh smelling, nutrient rich organic matter. That end product looks like soil. But it isn't. Instead, it's compost, or humus.

Humus may or may not be present in appropriate levels in soil. There's no such thing as normal or typical soil. But there's such a thing as healthier or unhealthier soil. The healthier soil has appropriate levels of nutrients and humus, and adequate amounts of air and water pore space. The unhealthier soil has neither.