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.