The subjects of sampling frequency
and dynamic range
are important for any sampled time domain signal. In the CD audio format, these are set in stone and cannot be changed. I think these issues give some credence to the claims that vinyl is better than CDs
According to the Nyquist Theorem, to capture a time domain signal in discrete form and reconstruct it from these samples without loss, the rate of sampling must be 2 times the highest frequency present in the time domain signal. CD audio is sampled at 44100 Hz. Therefore, a audio signal could have frequencies up to to 22050 Hz and be captured perfectly in sampled form. This signal can then be reconstructed mathematically using a series of weighted sinc functions. I won't go into the details, but I really mean perfectly (you audiophiles can argue this point all day long but you'll be wrong). The Nyquist Theorem is discussed in excessive detail in most signal processing textbooks.
There are two problems with a fixed sampling rate though, higher frequencies present in the audio signal and the electronic system used to recreate the signal from its samples. Sometimes there are frequencies higher than 22050 Hz in audio; the lucky among us may be able to hear them. When a signal like this is sampled, these frequencies which are higher than the Nyquist Frequency are aliased and wrap around in the frequency domain, smearing together with other, lower frequencies. This causes distortion. Secondly, even thought you can mathematically reproduce the signal with sinc functions and so on, the actual electronics (your CD player, using sample and holds, filters, etc) don't do it perfectly. Add in more distortion.
Dynamic range is another issue with CD Audio. It's 16 bit, so the samples themselves can only occupy a discrete value between and 65535. The entire set of waveforms in an audio signal (parts in an orchestra, for example) has to be sampled and fit to a value in that range. Measurable sampling noise is a result of this operation. This is yet another source of nastiness in the reproduced audio. And think about it, is 2^16 values really enough to capture all the ranges in full bodied sound?
In my opinion, CD audio is very good but is probably limited for some types of music. When an audiophile says they can tell the difference between vinyl and a CD, they probably can. The super-CD format (22 bits I believe) is far better in terms of dynamic range and sampling. This may be able to fool even the best ear.