I haven't read every comment in here so forgive me if I am restating some things.
But a few facts:
Vinyl's dynamic range is about 80dB. 16-bit LPCM (CD Audio) is about 96.7dB. Just for reference, dB is a logarithmic scale. Every 3 decibels represents a DOUBLING of wave power.
AAC is a perceptual coding algorithm, so what one has to realize is that there's ways to reduce the amount of data needed to accurately reconstruct the same original signal. In the old days, ADPCM (adaptive delta pulse code modulation) achieved this by storing only the change in amplitude at each quantization interval, rather than the absolute value, this reduces the data needed to reproduce the signal losslessly. AAC has many more advanced methods of reducing data requirements, enough that the Audio Engineering Society has found 128 Kbps AAC sufficiently indistinguishable from 16 bit LPCM in true double blind tests (in other words, not internet tests).
Remember that the analogue signal is always reconstructed before playback. You can't "listen" to a digital pulse code modulated stream. It's in the reconstruction of signal where errors occur, heavily dependent upon the size of the sample & hold buffer.
Some of the other things that audiophiles (read: People with ZERO fundamental knowledge of digital recording, mixing and mastering) piss and moan about, like jitter, are absolutely nonsensical arguments because internal reclocking of the signal has been around in digital playback systems since at least 1985. You don't need to buy a super duper high end DAC to have this. Computer quartz clock synch is even better than blackburst generators that code SMPTE.
The sizable difference is not between MP3 or AAC and 16-bit linear PCM. Not even... because both those perceptual coding algorithms can, at 320kbps and 256kpbs, carry enough data to, with the perceptual coding methods used, reconstruct a signal that is indiscernible from the original audio to the average listener. Why? Well because at 16 bits stereo 44.1kHz there's only about 65,536 possible amplitude values per quantization interval (2 to the 16th power) in CD audio. So CD audio is itself quite limited to begin with.
HOWEVER, compare this to 24-bit Linear PCM. Here's where 16-bit CD audio is pathetic. 24-bit Linear PCM has 2 to the 24th power, or 16.78 MILLION possible amplitude values per quantization interval. This translates to a dynamic range of about 140dB.
But if you are listening to music mastered predominantly in the last twenty years, this is all immaterial, since the average loudness (LeqA) of a track has been jacked up from around -12 to -18dBFS (decibels full scale) to -9dBFS in the 90s and even as high as -4dBFS today.
Techniques such as amplitude pumping, brick wall filters and third party plugins such as FGX have basically completely wasted the superior fidelity and dynamic range of the digital medium. Everything is cranked to 11, and that, more than anything else, is why recordings sound so horrible.
Pick up a copy of Ahmad Jamal's Awakenings from the early 70s, or anything mastered by Bruce Swedien (including Michael Jackson's Off The Wall or Chic's "Le Freak")... and compare. Awakenings has an average loudness of -22dBFS, much closer to the dynamics of a certified Dolby Digital soundtrack (-27dBFS is Dolby Labs' Licensing Corporations stated standard).
You can take these older recordings and crank the volume on any system, even a crappy car stereo, and they sound fantastic. But newer recordings are mastered near clipping levels. They're garbage to begin with.
For further interest, a while ago I did a commentary on the so-called Loudness Wars here:
http://www.cinemalogue.com/2010/12/04/the-loudness-wars/
It's a two part video editorial that breaks down the problem visually with audio examples.