Yes. Well...it's not me saying it, it's Nyquist and Shannon. And in your sentence, you can replace "that much better" with "any better at all".
Check out this oscilloscope view of a square waveform (produced by a synth). It's only two values, but when I run it through a low-pass filter, it ends up being a sine wave:
Before I cut off the high frequencies, you can see all the higher frequency harmonics in the EQ Analyzer, that are responsible for those angled edges in the waveform, that give you a square wave. Now imagine that those are above 20 kHz, meaning you can't hear them. Once you run that reconstructed signal into a low pass filter at the right frequency, you get a sine wave at 20 kHz, the limit of human hearing. Meaning you got rid of anything unnecessary and end up with the same smooth waveform you had originally recorded.