This reminds me of something one of my psychology professors said to me: "the body needs rest, but the brain needs sleep" (it's a common enough saying, but that's where I heard it). And this is pretty well borne out by extensive research in a variety of fields.
First, is sleep a habit or addiction that can be kicked? Well, people have managed this, actually. And the results are quite clear: severely reduced performance, PTSD, hallucinations, psychosis, and eventually catatonic states. The military and old psychiatric studies do a thorough job of cataloguing the more severe end of the spectrum wit real world examples.
In common parlance, lack of sleep can be attained, and it universally drives people stupid right up until it drives them insane. This is the result not just of sleep-deprivation torture techniques, but with pharmaceuticals that interfere with the drive or ability to fall asleep. There are naturally occurring illnesses that make people unable to sleep, and these all end the same way: if a person does not sleep they lose their ability to function first effectively, then rationally, and finally one cannot function at all.
In this way, sleep is like eating, and far harder to find an alternative. Some research I recall studied meditation and some methods to induce a sleep-like state that partially reduced the need for sleep - but it was a minor reduction that did not even hint at the possibility of elimination.
But why do we need to sleep? Well, not being a sleep scientist myself, I defer to a nice Harvard write-up that says, in short "we don't really know for sure." But there are plenty of theories supported by research, and one of the most recent (and perhaps critical) is the role in brain plasticity:
One of the most recent and compelling explanations for why we sleep is
based on findings that sleep is correlated to changes in the structure
and organization of the brain. This phenomenon, known as brain
plasticity, is not entirely understood, but its connection to sleep
has several critical implications. It is becoming clear, for example,
that sleep plays a critical role in brain development in infants and
young children. Infants spend about 13 to 14 hours per day sleeping,
and about half of that time is spent in REM sleep, the stage in which
most dreams occur. A link between sleep and brain plasticity is
becoming clear in adults as well. This is seen in the effect that
sleep and sleep deprivation have on people's ability to learn and
perform a variety of tasks.
As to why the symptoms you experienced seem similar to drug withdrawal, this is partly a coincidence; there are many illnesses and body states that produce the exact same symptoms - ask anyone who's experienced "abdominal pain". This is actually how medicine use to work up until the modern age, where illness and disease was categorized based on symptoms. It just 'feels' like a good system, but this is also why looking up any system online will indicate you might have cancer; a fever has dozens of common causes, twitches even more, and pain has countless more.
As an aside, this is also a major area of growth in the clinical psychology/psychiatry field, where someday in the future we may learn enough about mental illness to no longer define conditions according to systems but by underlying causes and treatments. But I digress...
There is also partial similarity due to underlying physiology/psychology. Pain, for instance, has an extremely strong mental component (some would argue that it is the largest component on day-to-day pain), but also sleep seems to have a role in bodily maintenance tasks like tissue/muscle repair as well. So if you aren't sleeping you are miserable, and that makes pain worse and seem to emanate from everywhere, and if your body is repairing muscles at a lower rate than normal then you are experiencing a genuine increase in communicated muscle/tissue pain to make things even worse.
Finally, I would note that sleep is an addiction - but so is breathing. Just because it is an addiction doesn't mean it's something you can - or at least should - try to defeat :)