While reading Richard Hamming's The Art of Doing Science and Engineering, I came across a section that described a fundamental advantage digital (or discretized) systems had over analog (or continuous) ones:
In continuous signaling (transmission) you often have to amplify the signal to compensate for natural losses along the way. Any error made at one stage, before or during amplification, is naturally amplified by the next stage.
Compare this to discrete signaling. At each stage we do not amplify the signal, but rather we use the incoming pulse to gate, or not, a standard source of pulses; we actually use repeaters, not amplifiers. Noise introduced at one spot, if not too much to make the pulse detection wrong at the next repeater, is automatically removed. Thus with remarkable fidelity we can transmit a voice signal if we use digital signaling, and furthermore the equipment need not be built extremely accurately. We can use, if necessary, error detecting and error correcting codes to further defeat the noise.
Analog systems transmit information through some physical property of the medium - the amplitude or frequency of waves for transmission through space, the grooves of a vinyl record in case of transmission through time (storage).
Digital systems are different - the information to be transmitted is first converted to a digital (discretized) format, then transmitted through the physical properties of the medium. The receiver translates the physical properties back into the discretized format.
This additional layer of discretization allows us to overcome energy loss and noise that plague analog signals. A digital receiver can afford the physical signal having some noise - as long the signal is accurate enough to distinguish a one from a zero we can translate it back to digital form without error. There's also less noise overall because we don't amplify noise each time we re-transmit it. Each repeater can decode the physical signal back into discrete form, then encode the discrete form again and send that on to the next receiver without any added noise. We can further reduce error by using error-correcting codes.
To use an analogy with human language: boosting an analog signal is like hearing someone say something to you in a language you don't understand, and then transmitting it to someone else. Since you don't really understand the language, it's likely you might make an error in the process of transmission - you have to worry about every inflection and detail of pronounciation being correct, because you don't know what matters and what doesn't.
A digital message is more like someone telling you something in English (or any other language you understand). It easy to pass the core message on without errors. You can also proofread the message (i.e. use error-correcting codes) to ensure it makes sense. Digital systems take a physical signal (someone's voice in our analogy) and digest it into an discrete representation (the message they want to communicate) that can be understood and reasoned about. This allows them to retransmit the same message again with fewer errors.