Dialogue is some part of conversation, but look:
Well, sometimes they can be synonyms. (probably they are, you can use both words, but sometimes they look weird in sentence, it depends on context).
But, I wanna add that is only my point of view and english is not my first language, but in my mother tongue, words are the same as in english
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But if I say, "Our dialogue was useful," I mean my conversation with only one other person.
Dialogue is between two persons. Most part of the dialogue includes Questions and answers, requests and information. Dialogue is a conversational exchange between two people. Dialogue is for purpose.
Conversation is talking between two or more persons. In talking you can say everything, even you can joke, you can tell a story, you can laugh. Conversation is informal interchange of thoughts, information, etc., by spoken words; oral communication between persons; talk; colloquy.
Conversation is interactive, more-or-less spontaneous, communication between two or more conversants. Interactivity occurs because contributions to a conversation are response reactions to what has previously been said. Spontaneity occurs because a conversation must proceed, to some extent, and in some way, unpredictably.
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Apparently, Conversation is a form of interactive, spontaneous communication between two or more people who are following rules of etiquette. While dialogue is for (a) purpose. Being a Voice Recognition Design Engineer since 1985 these two words; converation and dialogue and their meaning is important to understand algorithmically as a programmable experience, and probably included in systems like Apple's Siri and IBM's Watson. There are other important Cloud organizations, companies, DOD, DARPA, IBM Watson Research Department, Carnegie Mellons, Kurzwild, Jim Baker (Dragon - Nuance), NASA, AVIOS (American Voice Input/output Society), which was headed and formed by Mr. Leon Lerman of Lockheed Missile and Space, now Lockheed Martin, and the AAAI. There are many other participants in the exploding voice recognition industry, and can be located on the Internet and www. Remember that this industry is very complex, which is the nature of any measured vibration, especially human speech languaging, which cannot be changed once spoken. I hope this helps, and may be more information than you requested, but to paraphrase;
Dialogue is a conversational exchange between two people. Dialogue is for purpose. Conversation is talking between two or more persons. In talking you can.
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Context has a lot to do with meaning, or semantics. In fact Context Menu Switching is accomplished by using algorithms created for the task, and they are working rather well in both Apple's Siri voice recognition voice command software, and IBM's Watson. Both systems are referred to by me as being, "Turn Key", they work immediately, and there are many other manufacturers involved and using voice technology today as you already know. With Apple's announcement recently voice command voice recognition systems are, or soon will be ubiqutous. Thank you Arthur C. Clark, 2001 A Space Odyssey, and all the many corporations and companies that paid for the historical science fiction movie. It is the most important event for the field of voice technology and the current adaption by everybody, including all the major automobile manufacturers, cell phone manufacturers, Smart Televisions, and the list is getting longer by the second. Human speech is the most natural way to access technological devices of any size, period. We were born to communicate, and speaking is the most natural way, while the QWERTY keyboard, mouse, and touch screen is nice and actually fun, but they must be learned (keyboard), mouse (movement and settings), and Touch Screen use. I hope you find this helpful. Finally, I worked for The International Trade Division of the Parker Pen Company, Janesville, Wisconsin when 2001 A Space Odyssey was released. Parker Pen was a major funding source for the movie, and I used the many advertising promotional material and programs, including product (the T-!) specially developed, promoted and sold to the consumer domestically, and international distributors thoughout the world. Parker Pen was well known, and world wide then. It still is with it's headquarters in the United Kingdom, and it has diversified itself, successfully. A little corporate insider news from many years ago.
Tool making and language creation and use are the two most important successes humans have achieved.