Dance and gymnastics do have many similar elements. Those who have done gymnastics will find they have an advantage in dance when it comes to flexibility, strength, leaps, jumps, turns, pointed toes and so on. Also your gymnastics tumbling skills can be very valuable in dance. Many dance teacher will put skills like cartwheels, handstands, rolls, walkovers, back handsprings, back tucks and aerials into their routines.
However, dance is like gymnastics in that it is much more difficult to excel if you do not start at an early age. There are many, many different styles of dance. Kids who dance are not like gymnasts they don't just do one style. They need to learn all the styles if they want to make it successfully as dancers. Main Styles include classical ballet,national, jazz, lyrical, contemporary, hip hop, tap, clogging, musical theatre and tumbling/acrobatics. If you want to dance seriously you will find they do as many hours of dance as gymnasts do.
At the competitive level dance is a lot more expensive than gymnastics too. You may have to may for each class individually and every routine needs a costume. Costumes range from $80-$400 each and expect to buy several every year. Every style needs a special pair of shoes which cost $50-$150.