At the blog Conflict Antiquities, where Sam Hardy regularly tracks the illicit traffic in looted and smuggled antiquities from war zones, he makes a major point: the videos show the Islamic State destroying large statues that are too large to smuggle out and sell; smuggled artworks and artifacts are a major source of revenue for ISIS. He notes:
There is no doubt that the Islamic State is profiting from the illicit trade in antiquities. Although the criminals have destroyed some ancient artefacts, the have also destroyed a lot of modern reproductions – as is visible, for example, around 00h03m58s. All this video really shows is that they are willing to destroy things that they can’t ship out and sell off.
A post by Lynda Albertson for the Association for Research into Crimes against Art has some "before" photos from a UNESCO report, identifying the gallery shown in the photo.
An AP story by Sinan Salaheddin quotes a professor from the Archaeological College of Mosul as saying that most of the artifacts are genuine and not replica; the story also confirms that the sites shown are the Mosul Museum and the Nergal Gate in the ruins of Nineveh.
Another heritage preservation website, Gates of Nineveh, has not yet posted on the video, but you can find a summary of earlier destruction here.