An analog watch is what we think of as the traditional watch, where time is indicated with the use of hands sweeping around a dial. Most watches and clocks are analog. The reference to an analog watch is a retronym, a term created after an object or concept has been introduced, to differentiate it and the newer object.
Thus the reference to an analog watch was commonly used after the production of the first digital watch display in 1972 by The Hamilton Watch Co of Lancaster, Pennsylvania.
The term analog or analogue refers to the representation of another concept or object, while preserving the main perceptual features of whatever is being presented for the physical stimuli in the environment. Thus the movements of the hands of an analogue watch are analogous to the passage of time. Many watches such as the Movado Museum Watch Collections have no numbers at all, and Franck Muller Crazy Hour Watch has the numbers in a haphazard, non-sequential display. Franck Muller's watch is extremely popular proving the fact that a watch with hands is truly analogues and does not even require symbolic numbers or numbers in the correct place.
A digital watch with its numbers is more of a symbolic representation of time. The numbers have been chosen arbitrarily to stand for something and does not perceptually resemble whatever is being represented. The numbers on a digital watch such as a "2" is a symbol for the concept of twoness and represents the quantity of two, but nothing about the symbol would suggest the meaning.
Watches featuring moon phases and Christiaan van der Klaauw's astrolobes are indeed an excellent miniature representation of the real heavens above, whereas Gerald Genta's retrograde and jumping hour watches is a combination of both symbolic and analog time representation. The retrograde aspect of the watch does represent the passage of time albeit on a liner or curvilinear scale whereas the jumping hours are completely symbolic and have no concrete representation to the actual flow of time.
In recent years, the amount of different representations of time have skyrocketed. Watch companies are unveiling new and innovative ideas, some of which completely restructure our perception of time display, such as Urwerk's masterpieces or Romaine Jerome's watches.
Time is abstract, and yet the most real. The possibilities in representing the passage of time are only limited by the imaginations of the human mind.