Vinyl contains the pure sine wave of an original analog recording (think studios and master tape). All digital copies are stepped sine waves, so they lose or distort some information. Digital quality depends on the analog-to-digital conversion of master tapes to CD/DVD.
Good analogy there, although a bit inaccurate. Analog recording is a complete record of the recorded event, in as much as the physical and electrical limitations of the sound producing device and, subsequently, the recording device, allows. To explain what is meant by those limitations, let's say the sum of music produced by a band in a recording studio is M. They use cables, microphones, pre-amplifiers, amplifiers, mixers, etc all of which take away something and add something from/to M. So we have [M - x], which is recorded by a recording device. The recording device is not perfect so it also subtracts and adds its own things, say, y. So you have [M - (x + y)]. Assuming [M - (x + y)] is what gets burned to CD/LP, your playback system will invariably add/subtract to/from the music and therefore what you hear is [M - (x + y + z)].
Music in most forms is not pure sine wave. A pure sine wave is very boring and piercing to listen to (and so used for calibrating equipments and audio-video transmission links). Music is a combination of lots of harmonics.
A digital recording is a sample of the recorded event, sampled 44100 times every second or Hertz (in case of CD recording).
I think most modern audio is digitally recorded using computers rather than old analog studios that had huge multichannel boxes with hundreds of knobs and sliders. The computers at work at such digital recording studios must be using algorithms at much higher bit rates so that a pure sine wave is generated than a stepped one.
As explained above, no matter how high the sampling rate, a sampled signal is always a subset of the original signal. "Pure sine wave" will not be generated even if the computer is very powerful or fast. Yes, the filtering algorithms can smooth out the waveform to closely approximate the original recorded event. So higher sampling rate is always better while recording.