In my time I have had the chance to work extensively with both adults and children. It has led to many interesting experiences and observations on different learning styles and how to teach them.
The main difference I noticed in adults versus children? Resistance to learning.
Children are literally learning machines. They are hardwired for it. This means that a child can learn and improve simply from exposure regardless of any desire to learn or understanding of the process. Let me repeat that:
"A child can learn something without wanting to learn it or understand it".
An unfair advantage maybe? Not necessarily. Being able to absorb the environment around oneself comes with both benefits and pitfalls. Bad habits and behaviors can easily be acquired. Once acquired, they stay for good.
After a child goes through a certain amount of development and experience, this is where the mindset changes and the line is drawn. Rather than learn and process new information as something novel, an adult is more likely to draw on past experience and knowledge for comparison. This creates a certain close-mindedness, but even more so a measure of limits.
Of course none of this is universal. There are some childish adults out there as well as some mature children.
I think one of the most important qualities to instill in children you are teaching is to have them acknowledge who they are ultimately working for: themselves. Often children are taught with the soldier mentality, not sure why or what they are really training for. Self-motivation is what we are looking to build, otherwise they are just as likely to quit.
As a teacher of adults it is vital to know the "hacks" that overcome learning resistance. This can take a bit of problem-solving as every student is unique. The method of having everyone do the same cookie-cutter program is not enough to break through most peoples' individual barriers. Ultimately what I try to teach is self-reliance. Learn a principle and how to apply it to yourself.
Remember that regardless of who you teach, you must set the example.