A band plays instruments. That is, the members who are playing instruments are considered part of “the band”.
A group is a collection of performers who perform together regardless of who is playing the instruments. Think the average boy band who perform in the spotlights accompanied by a group of faceless musicians who might not even be on the stage. These people could change from album to album or even show to show and no one would notice.
A project is a person or group of people who create musical pieces or songs without the structure of a band or group. One person might play an instrument, or they all might, or all people might play whatever, and it’s all mixed in with stock audio and licensed pieces. Think Enigma or basically any New Age or Dubstep project.