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The intelligence in an analog signal is in the shape of the signal - the amplitude, the waveshape, and the frequency. Anything that distorts the shape of the signal degrades the information. Nature introduces "noise" mostly in the form of amplitude changes - voltage spikes from static, sparks in machinery, automobile ignitions, lightning, etc.
In a digital signal, the intelligence is in the edges of the pulses. As long as the edges of the pulse arrive within a certain time window, the information is still there - regardless of changes in amplitude, waveshape, or in some cases even the pulse frequency.
If you sample your data in the center of a bit, the bit edges can move a little and you still get the same logical value. If bit edges move too much to the center of a bit due to noise, jitter, etc, you might get an error.
In the analog system, information is usually not discrete, it is continuous. For instance, if the signal is coded by the amplitude of the signal, and you've measured very small (1 nV) change of the amplitude, you consider this change as an information. Then any noise, which always exists in the analog world, will be considered as a noise of information.
In the digital world, the information itself is not continuous, it is discrete: represented by integer numbers (limited amount of numbers). To transmit digital information, these numbers are coded into some analog (continuous) values, such as amplitude, phase, or frequency, which can be noisy as well. Being decoded back to discrete values, the analog noise can easily eliminated (since we know what discrete levels of digital signal to expect), so that received signal is absolutely the same as transmitted.
In other words, the amount of information in the digital signal is already limited because of discretization, and this limited information can be easily separated from the noise. The analog signal can not (generally) be separated from the noise, which may have the same properties and spectrum as a signal of interest.
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