EDIT: I was writing this when DHW posted, he of course said the same thing more compactly!
In several ways:
1. The Ameritage system uses an old fashioned plaster of paris plug that one wets, and it sits in a plastic box that you put in the compartment in the Ameritage case wet to add humidity. These have been around at the very least since I was a little kid (and I am OLD). In the old days, players stuck them in the accessory compartment of the case, and held the accessory compartment open by sticking something in the compartment door to prevent it from fully closing. In the Ameritage case system, it simply sits near the headstock.
2. The plaster of paris plug's release of moisture is confined to the part of the case you put the plastic box in. Since it isn't put near the body of the guitar by design, it doesn't humidify the body of the guitar. Simply put, there isn't enough room in a guitar case to circulate the humidity to that part of the guitar, because the top of the case, the padding and lining sit against the guitar top and interfere with that.
In fact, the types of humidifiers where you'd wet a sponge that sits inside a rubber tube, or wet a sponge that sits on the strings over the sound hole, were designed to solve that very problem. The Ameritage system doesn't address that at all.
The Planet Waves system puts separate humidifiers inside the guitar and in a cloth bag near the headstock to address both areas of the guitar. It still can't do all that much for the neck, unless the case has enough air space.
3. The plaster plug in a box with holes in it is a solution that can leak, and simply releases moisture regardless of the RH in the room. In other words, it releases moisture, period. It doesn't regulate it. Some days it needs to be wet, and some days not, but it requires daily checking, then re-wetting or removal from the case.
4. The Planet Waves system maintains the same humidity no matter what by chemical action, and it's going to be 45-50% RH regardless of conditions outside the case, every day, until it hardens, which takes a couple of months. So the case only needs to be checked periodically. That's because not only is the design a hell of a lot more space-age, the packet is also capable of absorbing moisture.
5. Hygrometers - even digital ones - need to be calibrated in order to be accurate. This is done by putting them in a plastic bag of salt, sealing the bag until they are in a completely dry environment, and then going through a recalibration procedure. Having familiarized myself with this process, I can honestly say that it's a complete pain in the rear. The Ameritage system uses a little inexpensive digital hygrometer that will only give you a rough idea of the RH in one part of the case,
6. The Planet Waves system is so bonehead simple, and so maintenance free, that no one can think it's a pain. There is no daily checking. There is no soaking a plaster plug in the sink every day until it absorbs water. There is no worry about over or under-humidification. You put the three bags into the pouches, and you stick one in the case, and drape one over the strings of the guitar inside the sound hole. No leaking. No musty smell. No issues whatsoever. It's really amazing.
7. The PW system actually works! Unlike the others I've tried, from the plaster plugs in a box, to sound hole humidifiers, to room humidification, to every other PITA product, it's obvious that the guitar returns to its factory form after a couple of days, you can literally see it.
Do what you like, of course, but my guitars will never be without this product again. It's brilliant. I even use it in my electric guitar cases to keep the fingerboards and necks from getting too dry (I stick one packet inside the case near the neck or headstock). The beauty is that it also keeps the guitars from becoming too wet. Genius!
Thank you CoreyT for bringing this to my attention!!