The downloaded updates are just deltas from the previous release of the OS. These updates are assembled into (more or less) a "new OS image" on-device and then installed in the same manner a monolithic OS install is done. Since the device needs to make a copy of the entire OS image, you need a ton of free space.
Installing the "full" OS update from iTunes shouldn't require quite the large amount of space on device, since tethered to iTunes the existing OS image can just be replaced at once since the iTunes download is the full OS image and not just a diff. The diff saves on network bandwidth at the cost of large amounts of scratch space on the device.
