You're initial suspicions are correct. It is basically a big balloon. Here is why.
Although information is sparse, let's establish one thing: Spartan was just a carrier platform. It's sort of an interesting platform in that it has absolutely zero command and control. After release it executes its mission, is recovered, and the results are pulled off it after it is on the ground.
Spartan was just used to carry the inflatable antenna experiment. Antennas in space generally have an incentive to be physically large. Roughly speaking, a physically larger antenna will have higher gain. Gain is a phenomenon of antenna design. An ideal isotropic antenna radiates power equally in all directions. For a spacecraft communicating with Earth this is incredibly poor. Even a simple antenna design that radiates energy only in the 180 degrees toward the Earth would be twice as effective as an ideal antenna. You could use a poor antenna with a powerful transmitter, but this requires larger solar panels and a bigger battery. This increase the payload launch weight even more.
Real antennas with real gain have a huge number of variables to deal with. A television satellite antenna is often designed to radiate energy only towards the part of the Earth where viewers are expected to be. A deep space probe uses a high gain antenna, but just points at the Earth. The apparent size of Earth is extremely small, but by keeping pointed at Earth it can radiate more energy back towards Earth. This raises the Signal-to-Noise ratio. The Shannon-Hartley theorem establishes that a higher Signal to Noise ratio enables more information to be communicated in the same communications channel. It doesn't matter if you're 10 feet from the receiver or 10 light years away, this theorem holds true.
High gain antennas onboard a satellite usually have a complicated system that very slowly expands a tightly packed high gain antenna to its final shape (usually a parabola) once the satellite is in orbit. The antenna is packed because it is physically large, but not very dense. Trying to launch a high gain antenna onboard a satellite that was already in its normal parabolic shape would require an unrealistically payload fairing on the final stage of the rocket. A failure of the deployment system can render the satellite useless. The Galileo spacecraft high gain antenna never deployed properly. It got stuck while trying to assume a parabolic shape. As a result the engineers opted to use the low-gain antenna for all communications.
You can find some more information about this type of antenna on one manufacturer's website. Unfurlable Mesh Reflector Antennas is the term that Harris Corporation uses to refer to their product. There are many suppliers, I just have seen one of their antennas in person before.
An inflatable antenna could be much simpler! A small gas generator could slowly raise the pressure inside the antenna until the antenna assumes the desired shape. I am assuming this experiment used a gas generator, I don't actually know this. for This gas generator is probably lighter than the mechanical means. For the same weight, a satellite could carry a much larger inflatable antenna. This is very important if you are developing a surveillance satellite. A surveillance satellite needs as much gain as possible to detect faint signals that are radiated from transmitters on Earth. One example case for this type of system is ECHELON.
The antenna you see unfurling in the images is a common parabolic design. You can think of it is as a cut out section of a sphere. It's no different than the small dishes used on peoples home to receive satellite televisionThere is more information here on Wikipedia about parabolic antennas.
So in theory inflatable antennas should be used by almost every communication satellite. What I can't find information on is the long term practicality of such an antenna. There are a number of issues with such an antenna. Primarily, micrometeorites and space debris could puncture the antenna's structure. A few tiny holes might not matter. Most satellite carry some sort of propellant anyways, so they could use some of that to keep the antenna inflated in the presence of small leaks. Eventually you'd have so many leaks that you couldn't keep it inflated no matter how much gas you have. So the actual lifespan of such an antenna might be much lower than the commercially available products on the market at present.