This is basically an attempt at marking up web content in hAtom or hNews (an extension of hAtom). The
entry-content class doesn't get assigned to the whole post but only to the body text, without title (which is
entry-title). So it isn't really a replacement for
the_post_class() because that gets assigned to the whole post. But for
entry-content to make sense, the article needs a class of
hentry. So using
the_post_class(), it would be
the_post_class('hentry') to include this class in the output.
And it has a very practical use: this markup has been established as kind of a standard for all those web services crawling and reformatting content, like Instapaper, Readability, Pocket etc. All those services use hAtom markup to identify the correct portions of an article. See Readability's Article Publishing Guidelines for a full example.