That thread describing the differences in design is somewhat pointless IMHO. An AC motor works with AC power, a DC motor works with DC power. If you only have AC power, you cannot use a DC motor without some sort of power conversion equipment, and vice versa. Chosing to add that extra expense then begs a different question altogether. If that's the case. then your question should have been "Why would I choose one type over another if I have the choice available to me?" But to provide anything useful by way of an answer or opinion, there are a lot of other pertinent questions about your application that need to be asked and answered before any further discussion of the operating differences makes reasonable sense.
The only other reason for asking such a noobish question is that you are a student asking us to do your homework for you. Shame on you if you are..
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The differences in "working" are as numerous as the plethora of variations in designs of each. Fundamental differences (excluding the universal ac/dc design) begin with the power source, as already stated. There are exceptions to this general statement, but, the speed of most modern AC motors is dependent upon the number of poles or fields wound on its stator (squirrel-cage rotor designs), and on the frequency of the line current. Thus, to get 3 different speeds out of an AC motor, there must be three different sets of windings (or, external variation of the electrical connections within). Thus, a three speed AC motor is by design not truly optimized for any one of its 3 speeds. (Trade-offs are made, to achieve the speed versatility.) DC motors, for the most part, are [by design] capable of variable speed control by varying the voltage applied to the armature (rotating element), and thus afford essentially 'infinite' speed control. Decades ago, the all-time great "beginners" book for learning electric motors was "Rosenberg's Electric Motors" ... a veritable 'Bible' on the subject (its what I, myself, began with). Lo-and-behold, I just did a google-it, and... voila! [a new edition]
There's a more fundamental difference in "working" (the wording of the original question divulges an even deeper lack of knowledge of the subject material).
Volumes of information and data exist on the subject of selecting the right type of motor for any given application. Not to expound any further, the analogy being made is that you might spend hours and hours deciding on a color to paint your ('whatever', here)... but if you don't know the first thing about paints and protective coatings, you'd better ask for some professional assistance before you grab a can of the "right color" off the shelf and begin brushing/rolling/spraying away...!