The main difference between hard and soft corals is that hard corals produce carbon calcium that creates a skeletal structure and soft corals do not create carbon calcium, so the skeletal structure doesn't exist. Soft coral looks like oceanic vegetation, but it is coral and does live within hard coral reefs.
Hard coral is the coral responsible for creating and building up coral reefs. As skeletal structures die out, living corals grow on top of the old skeletons. There are deep sea hard coral that do not contribute to reefs but do create their own small coral colonies.
Soft coral looks like trees, fans or flowing branches. While soft coral does not create a reef, it can grow its own soft coral colony of grasses and trees.