"Realism" is one of those lightning rod terms which cause very angry debates in the RPG world. This is because it's broad enough and subjective enough that it can be percieved in different ways, it's important to many if not most RPGs (depending on how you define it of course

), and due to these two reasons alone, people willfully misinterpret what is meant by the term (as so often happens in any discussion of gaming theory) and discussions of the concept consist of people talking past each other.
Another major problem with realism is that many more or less sincere interpretations of "realistic" games in the early days of RPGs went off on rather odd, and notoriously unsuccessful tangents, which were subsequently internalized by many gamers as the meaning of realism (i.e. complexity). So as a result it has a really bad name in role playing gaming, people think realism in combat rules means things like fatigue points or hit locations, and that realism in terms of a setting means something from Monty Python's Holy Grail.
Realism is important in gaming for the reasons a lot of people cited here upthread, in practical terms it essentially means internal consistency which contributes to verisimilitude, immersion, and other things quite a few gamers really like (though many hate the term). And contributes to non-gamers being able to get into RPGs without too much investment in learning entirely new ways of thinking.
But it is usually conflated for detail or complexity, due to games like Rolemaster or even GURPS. Somehow we lost the concept from wargames that a realistic game can also be abstract and simple. As a result, the explicit rejection of realism by many game designers including Gary Gygax, leaving AD&D a wierd amalgum of fairly well researched (but flawed) medieval background with high fantasy, low fantasy, and comic book themes.
In 3.5 we had the mixed blessing of returning to an idea of balance, but by then the 'realistic' basis was so far astray that a largely artificial system was created which was highly complex while having very little relation to reality. This actually makes it harder for non gamers to get into it, since the shared reality of real people is useless, instead you have to speak the made up "klingon" language of this entirely artificial world. I think this contributed to shrinking the RPG demographic even more than it was, which in turn increased the 'drift' of basing game world physics (in things like combat rules or historical settings) further into "klingon" land, alienating non gamers from the genre still further. Which I think some gamers actually really like. Now we have the further complication that some of the strange ideas of early RPGs have been successfully introduced to a much wider audience by MMORPG's, and we are now seeing a second retrenching of these themes.
But one of the biggest problems with rejecting realism utterly is that you do lose this common ground, and unless you have already bought into the strange ideas of RPG gaming (like the notion that a 10" knife is essentially a nuisance weapon, or falling off a cliff sort of tickles) then many of the fundamental notions of the game strike newcomers as absurd, and are a turn-off. (assuming they aren't already MMORPG gamers of course!)
Another major problem which we saw in 3.5 was that trying to make logical sense (or 'balance') of a fundamentally unrealistic system causes the same kind of complexity creep that the early misguided attempts to apply "realism" did.
Historically, things like weapons and armor, fighting styles and techniques balanced each other out in a rather elegant way. Trying to reinvent that from scratch leaves you down a road paved with with double bladed swords and spiked chains that eventually leads you into a level of geekdom most people find a little off-putting.
Of course dragons and flying carpets are unrealistic, but they exist in fairy tales, films, and novels which are a lot more broadly popular than RPG's are. It is the underlying fabric of interneal consistency which makes them stand out and seem magical in literature. In RPG's, if everything is magical, nothing is. Conversely, making the underlying reality convincing can really make the strange and magical seem
strange and
magic, such as in such widely popular games as Call of Cthulhu.
Genre based games like TOON, or Paranoia, or SpellJammer are perfectly sound, even "realistic" in their own way, since they are internally consistent and match expectations people in the broader culture (everyone who watched a cartoon, say) can predict at least to some extent. Much of the mush-mash of particularly fantasy RPGs suffers from a distinct lack of grounding in realism, and the urge to just throw it out the window entirely won't solve these problems.
Realism is just a tool, a means toward the end of having a fun game. Some games may not require it at all, but you shouldn't make "unrealistic" assumptions about what it actually
is and throw the baby out with the bathwater as a result
G.