For me personally,
Natural Beekeeping means doing it how they did fifty years ago. With smoke, using whatever style of hive you like, and doing the management how you like. It means jack shit today because natural means jack shit compared to permaculture. People call their shit natural and they use tons of synthetic pesticides and antibiotics, plastic, nasty shit in their smokers, etc..
Permaculture Beekeeping is far more advanced. It is permaculture principles used in designing the elements of your apiary/beekeeping system.
Basics first.
Type of honeybees. Catching swarms, baiting swarms and breeding from hives not treated with pesticides and antibiotics. That is the permaculture standard. The natural standard is buying commercially produced queens that come from bees treated regularly.
Style of Hives/Houses. For style of hive you can do whatever you like, but there can be no foundation, no wires, no plastic, no paint. I personally recommend langstroth or top bar hive designs, and even hollowed logs for the northern climates. All designs are acceptable, and I believe the more artistic and creative, the more permaculture-esque it will be. Langstroth is far more efficient in terms of honey production and colony replication. I would recommend top bar hives for those who can't lift very much weight.
Frames are wonderful, you don't have to use foundation or wires. You can make them without very expensive tools. You can design and shape the frames yourself. A frame is a frame, plastic foundation is another thing completely. Bees will draw their own comb in a frame neatly and efficiently. You don't have to extract the honey from frames in expensive equipment. You can do comb honey in a frame without wax foundation. You can build your own extractor for next to nothing with basic tools if you are resourceful.
Management. A reverence for the bees and all of nature. A thoughtful mind state when working the hives. Being careful not to crush any bees while working. Natural beekeeping doesn't value individual bee lives over time. Permaculture does. Smoke has to be very limited use only, and never for harvesting. I would only allow smoke for very invasive processes like feeding frames of honey and manipulating frames to raise queens and when making nucs. Otherwise it is a tool to be used for the safety of the bees and for people. Natural beekeeping allows nasty shit to be burned in the smoke, and the smoke is used liberally to make working the bees faster and more pleasant for the beekeeper.
Feed. In a permaculture apiary, planting specifically for the honeybees and pollinators in general is the difference. Taking all plants into consideration and listening to what the bees are wanting at different times of the year. This is how you feed honeybees, planting seeds or plants for pollinator forage. Natural beekeeping relies on what the farmer plants which usually isn't placing what the bees need high on the list.
Design of a system. It means using honeybees as one part of your system to feed and interact in many ways with other elements of the whole system. What crops you plant, how you provide
water and
shelter for the bees is part of the permaculture design.
It means not sending your bees to pollination, but bringing the forage to the bees doorstep. The bear-proof bee-hut makes it a permaculture apiary.
It is expressed as the knapweed honey in your tea or bacon cure, your royal jelly harvested when the hawthorn, chokecherry and service berries are in bloom. It is dandelion pollen harvested from your
polyculture, stock pastures. Its the wild areas left for predators and prey alike.
Doing what Paul advocates (catch/bait swarms, beehut, no foundation) is permaculture, feeding your bees gmo beet sugar water is considered natural.
I would like to emphasize that the design of the shape and style of the hive is not as important as doing what is best for your situation. It is not as important as what you build it out of, and how you manage it once it is established. The design of the hive structure will make the system as a whole, work for the better or worse.