The two words are not the same, and are not related.
You're right, MK, but as the Online Dictionary of Etymology points out in its
narc entry, "Sense and spelling tending to merge with older but unrelated
nark". However, I'm not so sure that the spelling shows much tendency to merge.
In 2006,
Cosmopolitan advised: "Don't be the workplace narc. Say something if you witness an illegal going-on, but it's smarter to bite your tongue when you spy a minor incident." Presumably, the "minor incidents" wouldn't involve narcotics.
And in the 2007 movie
Juno, the teenaged title character said, alluding to her pregnancy, "I asked my dad and Bren not to narc us out to your folks, so we should be safe."
Evidently,
narc is taking on the meaning of a snitch or informer, without necessarily relating to illegal drugs, or, indeed, anything illegal at all. So it overlaps with the meanings of the British
nark, but without generally changing its spelling. Similarly, as a verb, often with
out, it can mean to report on someone's behaviour, but not necessarily to the police. Norbert the Narc would be aghast!
Some dictionaries give
nark as an alternate spelling for
narc, but from what I can see, the
K spelling is seldom used in the US, although that spelling
is used in the earliest citation in the
OED's entry for
narc, a 1966 quote from Timothy Leary.
So, I conclude that Tom's initial post is valid.
Nark is very seldom seen in the US, so it should be rare - along with
narked,
narking and
narky. As for
narc, it was changed from common to rare last year, which I think is also correct, as that word is little used outside the US (and is perhaps a bit of a niche word in the US anyway).
I should also mention that a number of dictionaries identify some of the meanings of
nark - such as an annoying person or thing - as being mainly used in Australia and New Zealand. I'm not sure how true that is, but it isn't relevant to the question at hand.