Dynamic range compression and lossy file compression are completely different things. What's the difference?
Dynamic range compression squashes soft-to-loud volume shifts. This form of compression has been used by recording, mixing and mastering engineers for decades.
Other than bona-fide audiophile recordings, most of the music you hear has been dynamically compressed--which isn't necessarily a bad thing, as dynamic range compression adds punch, presence, and impact to music.
It's just that over the past decade or so the trend is to, so not only has music lost most of its natural soft-to-loud dynamics, but nuance and subtle detail are missing as well. The loud-all-the-time aesthetic is boring.
Recordings with less compression have lower (quieter) overall volume, so if you go from listening to maximally compressed contemporary recordings to something with less compression you need to turn up the volume to compensate for the difference.
As a consumer of music, you don't have the option of buying uncompressed music. If the engineers squashed the soft-to-loud dynamics out of the new Lady Gaga record there's no way of getting them back. Once sound is compressed, you can't decompress it. If you want to hear music with less compression, buy original pressings of 1960s or 1970s LPs. Yes, some of those will be compressed, but less than contemporary recordings.
On the other hand, you don't have to buy lossy compressed music, at least as long as CDs are made. I'm discussing lossy audio compression in this blog, but lossy compression is also used to compress video, streaming media, still images, telecommunications, etc. Lossy audio compression reduces the transmission bandwidth requirement of digital audio streams and the storage size of audio files.
For music, MP3s intentionally "lose" sounds that the MP3 software designers deemed to be inaudible. In the real world, the amount of compression and the applied bit rate determine how much audible sound is lost by compression. At 128kbps, it's easy to hear the losses, those MP3s sound harsh, grainy, and crude compared with CDs, which are completely uncompressed. Other forms of lossy compression include AAC, Dolby Digital, and DTS.
At 320 kbps, the difference between MP3 and CD sound is barely audible; you really have to listen carefully to hear it. Listening on a train, plane, bus, or car I doubt you could hear the difference. At home, played over a decent stereo system in a quiet room, the CD would sound a bit clearer.
And now the really bad news.
If you buy iTunes or Amazon MP3 files you're getting dynamically compressed, lossy compressed music. Congratulations, you're getting the worst of both forms of compression! You can avoid lossy compression by buying the CD, but chances are it will still be dynamically compressed.