https://en.wikipedia.org/wiki/Distortion#Harmonic_distortion
As the word "distortion" in that link implies, where audiophiles go wrong is confusing "nicer" with "better."
70s dub-reggae records and the like would undoubtably sound worse if they were recorded straight to digital and pressed to CD. Similarly, lots of 21st century music would lose its edge if it were printed to tape and pressed to vinyl.
Most of this confusion stems from audiophiles borrowing terms and ideas from music engineers and artists, much like the neverending stream of badness from managers/MBAs taking ideas and terms they don't understand from engineers.
If you are listening to music, it is none of your business to decide that it needs to be "warmer." The people who made the record spent a lot of time and effort making it exactly as warm as they wanted it to be. Pushing it through a tube amp to make it warmer before you listen to it is like retouching a Monet print in photoshop before you hang it on your wall. As you can imagine, I'm not a fan of the EQ settings on ipods either.
When optimising your playback system, go for maximum transparency, and stop there. If you don't like how the record sounds, listen to better records.
If you're making records OTOH, it is entirely proper for you to be worrying endlessly about "warmth". Buy up crazy 70s gear from ebay, have the track pressed to 60g vinyl at a boutique german cutting house, whatever it takes. Warmth is tricky :)
There's nothing preventing audio engineers from introducing harmonic distortion, wow, flutter, pops and ticks, reduced channel separation or equalizing a CD master differently.
There's nothing magical about vinyls and they are inferior to CDs in all possible ways.
Yes but if you want it to sound like vinyl you have to press it to vinyl. Introducing all those things in software will generally sound kinda crappy and it's fiddly. You can do it in hardware, but that's likely more effort than just sending it off to get an acetate pressed or something. I'm not campaigning for it, just trying to make the point that there are no rights and wrongs in a creative process.
And there is something magical about vinyl: the 1:1 correspondence between the physical medium and the stored sound means that DJing vinyl will always be a more satisfying experience for the DJ than any other playback technology. A turntable has almost no state to worry about: once can operate a pair of decks and a mixer blindfolded, by touch alone. The same is not (generally) true of traktor or CDJs. This promotes the "flow" state of mind, and therefore higher quality DJing.
This is the primary reason why vinyl still exists in 2013, and it's the only reason why countless twentysomethings like me have thousands of records stacked up in their bedrooms.
This is one important reason why the music on vinyl sounds nicer. Not because of the harmonics, but because of the mastering. The medium does not allow very high dynamics and you need to compromise.
Also, like I said above, it's about conveying the intention of the artist. While that recent metallica album sounded like shit and might have been better if it were mastered for vinyl, the wax pressing of the below-linked record sounds boring and lifeless because it lacks the downloadable version's obscene dynamic range:
Albums like Death Magnetic are particularly egregious examples of this, but I think that every CD suffers somewhat from this obsession with making the CD itself so damn loud. We have amplifiers for a reason.
The idea is is that if you can make a mix sound good through them, they'll satisfy as a general purpose sound product.
