I can't find any reference for exactly what goes on behind the scenes with an iOS app update, but for pretty much any program upgrade, the system shouldn't just delete the existing app then install the new one. It will unpack and install the new one, copy the settings, save games, etc. then remove the old one. Even if you only remove the app binaries (program code) prior to installing the new code and not the user data, you run the risk of getting a bad download or corrupt compression and not having a valid program for the user to run.
So, the reason you can't install is that iOS needs some scratch space to decompress the file, and will almost certainly have a large portion of both versions on disk at some point. And I believe iOS also needs a bit of breathing room, so I suspect the app updater program has fairly conservative limits on how much free space it needs when installing something.
Unless you don't have any saved games or other data that you want to keep, I would recommend not deleting then installing the new version, but rather try to free up some other space by deleting apps or media that you can replace.