Think of musical performances as an exam. The student has been studying, practicing, and learning for months, and a concert is the opportunity to take all of what they've learned, assemble it into a whole, then deliver it back to an audience. Since playing an instrument is both a mental and physical activity, proficiency must be demonstrated in both mental and physical applications. In other words, the only way to test students' ability to make music is to ask them to make music.
Performing in recitals goes another step in putting the completion of the "exam" solely on the student. They do not perform hidden in a concert band - they're up on stage by themselves or in a group of 2 to 4 or 5. They are the only ones who can perform their piece, and there's no one to act as a safety net. Success and failure are both real possibilities. For a few minutes, they have control of the room, and every time they succeed in delivering a solid performance, they come out a little more poised, more confident, and less fearful of an audience - all traits that will be helpful to them in the future.
In the studio, students don’t really have opportunity to hear their studio mates play. The most interaction they have are a few words exchanged as one student leaves and the other arrives. In studio recitals, students get to listen to their studio mates perform, and even get to play in small ensembles with them from time to time.
A performance is the natural culmination of the art that we musicians study. Art by definition results in a visible, tangible, or audible product which is the vessel of the artist’s expression. Even when we play pieces not written by ourselves, we still impart imagination and creativity to the performance. Music-making isn’t complete without witnessing the act of music-making. Unlike an ordinary exam, like math or history, performances have the artistic element. It's not enough for the notes and rhythms to be accurate - a computer can do that. But, the musician can put his or her own creativity into the reading of the music much like an actor does to a script. The music is made meaningful when there is tone, inflection, intensity of volume or articulation. The product of this preparation and personality is absolutely worth sharing with friends, family, and classmates.