Analog vs. Digital:
An analog format such as vinyl records are said to be more accurate to the source of the music than digital formats, which convert the music into a digital format, before being converted back to analog for reproduction.
Vinyl records provide a more rich, warm tone than digital formats. The reason for this better tone is because the grooves on records provide more musical data to your amplifier. When the stylus (needle) fits into the groove of the record, it vibrates in such a way that is reminiscent of the soundwaves present when the artists performed the song. When amplified, you get a reproduction of the original performance.
When the musical data is converted into a digital format, there are various "resolutions". To use photos as an example, it would be the difference between a 100x100 pixel picture, and a 1000x1000 pixel picture. For CDs, it is 16-bit. For DVD Audio up to 24-bit. Then, there is a sample rate, which is how often these photos are taken. For CDs, they happen 44,100 times per second. DVD Audio, up to 192,000 times per second. For SACDs, an amazing 2,000,000+ samples per second. You can see how the more advanced digital formats would sound better than CDs, which is how most people listen to digital music.
In the case of digital formats like MP3 and AAC, some of the higher frequencies get pulled out to save space, in addition, these formats are taken from a CD, and compressed, so they could never approach the sound quality of a CD. I expect digital music formats to continue to gain in popularity, and I also expect even better formats to come out, with new digital format remasters, which could sound much better than a CD (Or even DVD-A or SACD), have more customization options, and of course, undisputed portability. Are you listening, record companies?
How specifically vinyl sounds better:
Vinyl seems to reproduce the higher frequencies much better. On digital, high frequencies (voices, cymbals, drums, etc.) sound thinner, harsher, "swishy", simply not as good. On vinyl, these high frequencies sound more natural, less harsh, more "real". Also, very small high-pitched nuances shine through better, like for example when a bass player plucks a string.
Bass seems to sound fuller on vinyl as well. It sounds thicker, richer, more natural, more nuanced. It's nice. When listening to bass on vinyl, you'll know why so many DJs still use vinyl to this day. On digital, bass seems to be muddier, thinner...it has less impact.
It's true that vinyl does have some weaknesses, such as pops and hiss, but these are eliminated if the record is clean. The pops and hiss also add a nostalgic charm to the sound of the music. Some artists, especially hip hop, have incorporated this sound into their music.
If you love the sound of music, and want to hear something that has a better sound than CDs and MP3s, you should give vinyl a listen. Once you hear it, you'll become a fan too!