Hear the music as the artist or composer intended. A feel for the music and what the artist is trying to convey can only be realized on an instrument that is in tune.
A properly tuned piano helps "train the ear" of both player and listener. The harmonious relationship between two or more beautifully tuned notes or intervals (chords) played simultaneously is the essence of what music is all about! And don't think your ears don't have the potential to recognize it! A simple demonstration at the piano during your next piano tuning will convince you! My customers comment all the time when I sit down to start tuning. "Wow! I didn't realize how out of tune my piano was! If you never or rarely ever have your piano tuned you are never training your ear to recognize an out of tune condition.
We love to hear our children play an instrument, however, the daily grind and repetition of hearing the same songs over and over again wears on even the most enthusiastic parent. Worse yet, most of us are listening to our children practice on an out of tune instrument. A properly tuned piano won't keep your young student from hitting a wrong note but when they do play it wrong for the tenth consecutive time I promise you it won't sound as much like fingers on a chalkboard! And when they do get it right...? You'll appreciate their success much more on a properly tuned piano.
A properly tuned and maintained piano helps fuel your child's enthusiasm to sit down and practice. You're spending all that money on piano lessons, a little more to assure your piano is in tune and plays properly will reward you with a more eager to practice child.