CDs are digital recordings, whereas vinyl records are analog recordings. When an analog signal is captured digitally on a CD, it loses its wide range of frequencies. Therefore, the details of a performance aren't fully articulated in the digital snapshots on a CD. By definition, analog is natural sound, and vinyl records produce a less artificial sound reproduction. Vinyl's wide grooves accurately capture the waveforms of the original recording, but wear down after repeated plays and create more distortion in the playback.
A properly produced album will use different mastering techniques to accommodate their respective formats. Unless a recording has an alternate mastering job, the overall quality difference between the CD and vinyl will be minimal to a casual listener. Many modern CDs are mastered loudly in an attempt to stand out from others, but this results in an overcompressed recording that fatigues the ears. Because of their physical limitations, vinyl records don't suffer from compression like CDs do.
Aside from the large artwork and record, the vinyl format carries different sonic aesthetics than the CD. Vinyl records have warm, harmonic distortion and popping noises that are absent in CDs. During quieter parts of songs though, this static may overpower the music itself. Scratches on a CD may render it unplayable, while the scratches on a vinyl record may complement the nostalgic experience of spinning an LP.
Having a proper sound system to play your CDs or vinyl records through greatly affects the quality. A turntable with a worn-out needle won't be able to accurately convey the music within the grooves, just as a stereo with blown out speakers won't properly deliver the data on a CD. High-end audio equipment surely won't make a poor recording sound good, but it is indeed essential for attaining a high fidelity playback in both mediums.
Twenty-first century music fans may not mind lower quality recordings, and as a case study at Stanford University suggests, they may even prefer it. The seven-year study conducted by music professor Jonathan Berger found that an increasing number of students preferred playbacks with lower fidelity. Achieving high fidelity may not be a priority of all listeners, but for those keen on it, both CDs and vinyl mediums have unique characteristics to help you find your truer sound.