There are just so many different terms for different things, and it gets even more complicated when we use the same term for different things. Matt is on the right track, though. S-OFF = Nand unlocked. This doesn't mean you have a rooted ROM or can run root only apps. It is, however, "permanent" in the sense that you kind of have to purposefully remove it. If you rooted with unrevoked, you have to unroot with the S-ON tool. There is no other way to remove it, technically. If you rooted via the eng bootloader, you can lose S-OFF by upgrading hboot. This is why so many people lost root back before unrevoked was popular. In any case, if you have S-OFF, you gain the ability to flash unsigned images to all internal partitions, including flashing different radios to whatever versions you want. It also gives you the ability to make permanent "one-off" changes to "locked" partitions. I say locked in the sense that only signed images from HTC can normally change things there. With S-OFF, this is really all you gain. It's big, though, because it means the ability to flash ROMs. In my opinion, it's a misnomer to call this root. It's commonly referred to as full root, but it's really just unlocked nand. It has nothing to do with root level permissions. If you get S-OFF, and don't have a rooted ROM, you still won't be able to make changes to /system, for example. You need root level permissions to make changes there, as files are owned by root, plus you have to mount such partitions in rw mode, which only the root user can do. It would however, allow you to make changes outside of the Android OS. That's where recovery comes into play. S-OFF and recovery go hand in hand. One thing *allows* you to make changes (S-OFF), while the other actually *makes* the changes (recovery).
Those two things aside, what I would consider root in the true sense of the word is having the ability to run things as the root user and actually become the root user. This is more or less accurate to what you're referring to in a Linux/Unix system. All current temp root solutions are called temp root because they cannot keep the ROM rooted after a reboot. Why? because you can't make permanent changes to /system without it. You might be asking yourself, though, if S-OFF just unlocks nand, what does it have to do with root level permissions in a ROM? Well, try this - root with unrevoked, then flash the S-ON tool. Since it made all the changes it need while you had S-OFF to root the ROM, you should keep root permissions in the ROM even though you now have S-ON. Why? Because you had S-OFF at one point. Just remember this - there are three things to what we call "full root" - S-OFF, a rooted ROM, and a custom recovery. They are more or less independent, but you need S-OFF to get the rest even though you can get rid of S-OFF later, and keep the rest. You can even have a custom recovery with S-ON. It's sort of pointless, but you can certainly do it.