The main difference between the two is the type of bodies that will be shown during these competitions. For figure competitions, those who would join are expected to be muscular. They have to make sure that they will have the right muscles that will make them easy to recognize and become chosen as the best.
For bikini competitions, the women who will join the competition will have fewer muscles, although they are still expected to be toned. Expect that for figure competitions, the participants will do their best to gain more muscle. If you are older, a figure competition will be more ideal since people tend to lose body fat as they grow older.