Answer updated for 2020:
Both <object>
and <embed>
are included in the WHAT-WG HTML Living Standard (Sept 2020).
The object element can represent an external resource, which,
depending on the type of the resource, will either be treated as an
image, as a child browsing context, or as an external resource to be
processed by a plugin.
The embed element provides an integration point for an external (typically non-HTML) application or interactive content.
Are there advantages/disadvantages to using one tag vs. the other?
The opinion of Mozilla Developer Network (MDN) appears (albeit fairly subtly) to very marginally favour <object>
over <embed>
but, overwhelmingly, MDN wants to recommend that wherever you can, you avoid embedding external content entirely.
[...] you are unlikely to use these elements very much — Applets
haven't been used for years, Flash is no longer very popular, due to a
number of reasons (see The case against plugins, below), PDFs tend to
be better linked to than embedded, and other content such as images
and video have much better, easier elements to handle those. Plugins
and these embedding methods are really a legacy technology, and we are
mainly mentioning them in case you come across them in certain
circumstances like intranets, or enterprise projects.
Once upon a time, plugins were indispensable on the Web. Remember the
days when you had to install Adobe Flash Player just to watch a movie
online? And then you constantly got annoying alerts about updating
Flash Player and your Java Runtime Environment. Web technologies have
since grown much more robust, and those days are over. For virtually
all applications, it's time to stop delivering content that depends on
plugins and start taking advantage of Web technologies instead.
Source: https://developer.mozilla.org/en-US/docs/Learn/HTML/Multimedia_and_embedding/Other_embedding_technologies#The_%3Cembed%3E_and_%3Cobject%3E_elements
<embed>
is now officially a standard tag with HTML5, but you should anticipate at least some compatibility issues with older browser versions. – Vocational