Before we get into our deeply dodgy, pseudo-scientific investigation, let's establish exactly what we're talking about. MP3, AAC and the other main audio codecs were designed to reduce the amount of data required for a digital audio file to provide an accurate representation of an original recording, enabling these files to be sent over the low-bandwidth 1990s Internet or packed en-masse onto the earliest, low-capacity MP3 players. All use some form of lossy compression through perceptual coding - essentially, analysing the music streaming from CD and using a psycho-acoustic model to work out which bits will be marginal to or beyond the aural capabilities of most human ears. This information can then be reduced or even discarded, resulting in a smaller file.
Up to a point, it's a good approach, but it has limitations. First, much depends on the encoder being smart enough to work out which parts of the audio stream are crucial and which parts aren't, which is why you might hear a surprising difference in audio quality between MP3 tracks encoded at the same bit rate but using different encoders.
Secondly, there's a reasonable variation in the sensitivity of human ears. Those people who have naturally sensitive ears and/or who have trained their brains to recognise slight differences in audio quality (e.g. audio technicians, classical musicians) may spot flaws in MP3 tracks that most people wouldn't notice in a million years. Audiophiles take these things seriously, and when you've spent hundreds or thousands of pounds on Hi-Fi equipment, you aren't keen to make any compromises on their source material.
Finally, lossy encoding methods inevitably create certain audio artefacts. Instruments or sounds that have a short, fierce initial attack, like percussion, can be smeared or affected by 'pre-echo' (where the actual sound is preceded by its echo, with any recorded echo diminished). Some audiophiles claim that the simple fact that if any data is discarded, the result will always be a lower quality listening experience, particularly when high-end equipment is used.
This is where lossless formats come in. Lossless formats, such as Monkey's Audio, FLAC, Apple Lossless and WMA Lossless, are far less efficient when it comes to squeezing lots of audio data into a smaller package, but by retaining all the information they ensure that the compressed version is the equal of the original. Now, this needs to be kept in perspective. To hear some people talk about FLAC or Apple Lossless, you might think that we're talking about the ultimate in audio quality. However, for obvious reasons, a FLAC encode ripped from CD can only ever offer the same quality as the 44.1kHz, 16-bit Linear PCM original. While it's possible to purchase tracks in a 96kHz, 24-bit FLAC format from specialist stores, most FLAC files you'll produce or come across won't match, say, Super Audio CD (2822.4kHz, 1-bit DSD) or DVD Audio (48kHz, 24-bit) material for quality. Given that these failed physical formats were designed to overcome the perceived limitations of CD, it's worth giving this some thought.