The major difference is in the strength. The forging process yields preferential grain flow which improves strength (in one direction). As was mentioned, you start with a hot chunk of metal and progressively form it with dies until it is close to the final shape.

Billet (a fancy way to say machined from solid) parts are just machined from a chunk of metal. The material is homogeneous. The machining will cut across the grain, interrupting the pattern.

Depending upon how the initial chunk or material was formed, a machined from solid piece may be stronger than a cast piece, but a properly forged piece will be stronger than either.