Semantic tags simply carry meaning in themselves whereas div-tags don't. There was a campaign to use standardized attributes to convey meaning (the campaigns name escapes me at the moment), but that's just a hack.

Just an example, if you want to scrape a website for the first 100 characters of its articles, you can just look for <article>-tags. If the website used <div>-tags you would have to figure out how to identify the articles (perhaps the author used the class .article or .main_article, etc.), and you'd have to do it for every website you want to scrape.

Screen readers would be another example of software that profits from semantic tags.