There are a lot of people who would not consider dance a sport, including people who do it. There are differences: the performance for a dancer is usually pretty similar if they are in a stable show. There is rarely any sort of overt competition within a show, there is no trophy or big league game at the end of a season, but there are more similarities than these differences.
Like any sport, dance requires a lot of training and discipline. Most dancers start at a young age, often before they can start playing a “regular” sport, at 2, 3, or 4 years old. As a dancer’s childhood goes on, especially a child whose interest is in ballet, they begin to train for longer hours after school, sometimes quitting school altogether to focus on dance. Around age 11, most young ballerina’s are beginning Pointe lessons, in which they must buy a pair of wooden shoes that they will stand and dance on for hours. These shoes need to be replaced regularly.
By the time these dancers are in high school, many of them attend lessons for 5-6 hours every day after school, longer if they attend a performing arts school. Public do not usually have dance programs within their school system, so dancers must pay for lessons outside of school. Dancers may start to audition for companies or ballet schools when they get closer to 18. After that, dancers will be auditioning for jobs regularly, whether that is within a specific dance company or backup dancing or choreographing, there is rarely stability. Dancers often need to move to big cities, especially New York or Los Angeles in the US, in order to even be able to audition.
Unlike “regular” sports, there is no “off-season” for dancers. While other sports and athletes have time off between competitive seasons and games, dancers will most likely perform shows back to back for an entire year or more, with maybe one day off a week. Their bodies are not given the rest and recovery period that “regular” athletes are given because they are always performing and need to work in order to pay their bills, as they make significantly less money than traditional professional athletes. This can cause many injuries and problems for dancers.
Many dancers, although advised to sit out performances or practices, even encouraged to take a year off of dance, will refuse to or take those words lightly. According to Hopkins Medicine, the most common dance injuries include hip, ankle, knee, and foot injuries, as well as stress fractures all over their bodies, and are extremely prone to developing arthritis. Although dancers are a lower risk for ACL injuries, they tend to deal with chronic pain and swelling.
While traditional sports teams have medical professionals on the side of the field or rink, companies would be lucky to have the money to hire a professional. Dancers will often self-medicate and self-wrap their own injuries before going on stage. Often, if a dancer injures themselves during a show, they will take pain medication and wrap their injury inconspicuously and keep on dancing, as they must finish their show.
Dancing is both an art and a sport because not everyone can work that hard with that many permanent injuries and look so effortless.
Source: Hopkins Medicine
Paul McConnell 
Daniel Finegan 
Will Cunningham