Both hashtags and pushState are ways of changing the page and the URL using Javascript. pushState is the "legal" way, which changes the page's URL (i.e. twitter/posts/1 -> twitter/posts/2). hashtags technically don't change the page, they change the part of the URL after the hash (i.e. twitter/posts#1 -> twitter/posts#2).
hashtags are used because older browsers don't let you use pushState. pushState is used because it's more correct, and doesn't exploit a hacky loophole.
Both are used because they are faster than a whole new page request, and they let you share the post by cutting and pasting the URL in your address bar.
