It is going to be different for each school. You mentioned in the other thread that everyone learns the same skills. I would assume then that you will continue to work on the same techniques you did as a white belt.
I will answer your question for purple belt in my experience, because purple is the third belt (first after yellow) in my old school.
Purple Belts at my school learned the following:
- Kicking: New footwork (step-and-kick and step-behind side kick). New kick in the back kick. Jumping kicks (jumping front, jumping roundhouse, flying side kick). Students are expected to learn and memorize Sliding Kicks #1-5 and Jumping Kicks #1-3.
- Strikes: In addition to the basic punches we do in the white and yellow belt class, purple belts learn the chop-spin-chop combo.
- Forms: Adults will already know Kibon #1-3 and will learn Kibon #4. Kids will know Kibon #1 and will learn Kibon #2-3. Kibon #2 adds kicks to Kibon #1, Kibon #3 adds a lot more blocks to Kibon #1, and Kibon #4 combines 2 and 3 together.
- Self-Defense: Kids and Adults will both learn "Purple Belt Punch Defense #1-5", however the kids are learning the second set of punch defenses and adults learning the third (because kids start memorizing self-defense at Yellow and adults at White, same as forms). These are more complicated than the previous defenses. Adults will also learn "Straight Grab Defense #1-3", to contrast with the "Cross Grab Defense" series they had at White and Yellow.
Purple Belt is also when you get sparring gear. White and Yellow belts would only do non-contact sparring, in which you take turns moving backward and forward shadowboxing with an opponent just out of range. Purple belts get their sparring gear and do contact sparring, where you're allowed to strike the chestguard and you don't have to take turns or stay in your line anymore.
Another difference is that the Purple and Orange belt class will zip through content that we would do slowly in the White and Yellow Belt class. For example, going into Horse Stance takes a second to do in one motion instead of 5 seconds to do step-by-step. A drill using a roundhouse kick we can describe as "roundhouse kick" instead of "roundhouse kick: chamber, pivot, snap, recoil, and land."
A lot of people quit when they get their purple belt. Because all of a sudden they need to memorize kicks, and the new footwork is hard, and they think they already knew "the form" and now there's more to learn, and they go from being top dog in white and yellow to bottom of the barrel in purple and orange. It's a tough class to teach because you want to push the students, but also don't want them to get too far over their head.