The answer lies in the difference between analog and digital recordings. A vinyl record is an analog recording, and CDs and DVDs are digital recordings. Take a look at the graph below. Original sound is analog by definition. A digital recording takes snapshots of the analog signal at a certain rate (for CDs it is 44,100 times per second) and measures each snapshot with a certain accuracy (for CDs it is 16-bit, which means the value must be one of 65,536 possible values).
This means that, by definition, a digital recording is not capturing the complete sound wave. It is approximating it with a series of steps. Some sounds that have very quick transitions, such as a drum beat or a trumpet's tone, will be distorted because they change too quickly for the sample rate.
In your home stereo the CD or DVD player takes this digital recording and converts it to an analog signal, which is fed to your amplifier. The amplifier then raises the voltage of the signal to a level powerful enough to drive your speaker.
A vinyl record has a groove carved into it that mirrors the original sound's waveform. This means that no information is lost. The output of a record player is analog. It can be fed directly to your amplifier with no conversion.
This means that the waveforms from a vinyl recording can be much more accurate, and that can be heard in the richness of the sound. But there is a downside, any specks of dust or damage to the disc can be heard as noise or static. During quiet spots in songs this noise may be heard over the music. Digital recordings don't degrade over time, and if the digital recording contains silence, then there will be no noise.
From the graph you can see that CD quality audio does not do a very good job of replicating the original signal. The main ways to improve the quality of a digital recording are to increase the sampling rate and to increase the accuracy of the sampling.
The recording industry has a new standard for DVD audio discs that will greatly improve the sound quality. The table below lists the sampling rate and the accuracy for CD recordings, and the maximum sampling rate and accuracy for DVD recordings. DVDs can hold 74 minutes of music at their highest quality level. CDs can also hold 74 minutes of music. By lowering either the sampling rate or the accuracy, DVDs can hold more music. For instance a DVD can hold almost 7 hours of CD quality audio.