It’s not an either-or situation. HTML5 still has <div>s. It still has <span>s. The new tags are there to give you more expressive freedom and to standardize common elements. For instance, most pages have navigation bars, yet so far there was no standard for how those were marked up. If they’re clearly marked up as <nav>, programs can start to use this information productively. Search engines can ignore or parse them and assistive technology can help the user navigate around the site.
It also makes it easier for you to work with your code. <nav> is so much easier to spot than <div class="main nav foobar baz">.
The new tags are an addition, not an entirely new set replacing the old tags.