Unless the rabbi you ask your questions to is one of the higher-tier rabbis, you are likely not receiving any original information. When you ask him if something is permissible or prohibited he is merely summarizing the books for you. Were you to follow up and ask him to elaborate he would probably cite a chapter and paragraph in the Shulchan Aruch, or other such authoritative works of Jewish law.
To that end it is the same as Mi Yodeya. Here too when you ask a question you get a summary of the books. In fact, in that regard Mi Yodeya is better than an average rabbi because you will almost always receive the precise citation, often with a link to the page, where you can immediately look up the source. Additionally, when you ask the rabbi he might be answering you from memory, or might not have the precise source. As Mi Yodeya is "peer-reviewed" it is more likely that any mistakes will be caught than that your rabbi's mistake will be caught. Most importantly, when you ask your rabbi you are limited to one person's store of knowledge. Even if he is a great Torah scholar he is a human being with a limit to how much he can know. There are thousands of works of halacha and it is impossible for one person to even read all of them, let alone master them and remember them. And of course, an average rabbi knows much less than that. Most semicha programs cover parts of the 4 sections of Shulchan Aruch, with the Shulchan Aruch and the on-page commentaries being the main texts studied. There are so many other books of halakha, not to mention the thousands upon thousands of responsa that an average rabbi may have never studied. When you ask a question on Mi Yodeya, however, you can tap into the combined stores of knowledge of 1,000 users (or the hundred or so that actively participate). It is very likely that one of the users here will know a halachic source that your rabbi does not; it is very possible that you will get a more comprehensive answer here.
In short, if you have a very weak learning background, or you need an immediate answer, you may be better off asking your rabbi and trust him and hope that he gives you the correct answer. However, if you have a decent background in Jewish learning (i.e. you can comfortably navigate Jewish texts such as a Talmudic page or a responsum) and don't need an immediate answer I think you might be better off asking your question here. The answers you receive will direct you to relevant sources that you can then learn for yourself.
Essentially, I think the main reason that someone who learned in yeshivah for a number of years asks questions to a rabbi (aside from very serious questions) is that they don't have the time to learn through every sugya themselves, and/or they don't know offhand the specific sources to learn for a specific question. However, precisely these areas are where Mi Yodeya can have the advantage over an average rabbi.
Of course this does not take into account methodological issues. For instance if you ask a question here and you find that the Mishnah Berurah says one thing while the Aruch Hashulchan says the opposite, how will you decide which one to follow? If you ask a rabbi he will probably tell you which way to act. However, I don't think the average rabbi is especially well-trained in this either, and anyway if you are stuck with a methodological question after learning the sources you can ask that on Mi Yodeya too!
Of course if you don't feel comfortable relying on Mi Yodeya you can always ask your question here and then consult your rabbi afterwards.