...but the font is licensed to Microsoft (perhaps to others too?)
It is licensed to anyone who bought the license:
Times New Roman is a serif typeface commissioned by the British
newspaper The Times in 1931, created by Victor Lardent at the English
branch of Monotype. It was commissioned after Stanley Morison had
written an article criticizing The Times for being badly printed and
typographically antiquated. The font was supervised by Morison and
drawn by Victor Lardent, an artist from the advertising department of
The Times. Morison used an older font named Plantin as the basis for
his design, but made revisions for legibility and economy of space.
Morison's revision became known as Times New Roman and made its debut
in the 3 October 1932 issue of The Times newspaper. After one year,
the design was released for commercial sale.
But with respect to the glyphs they actually share, are these two
fonts any different at all?
In the links you provided, you are comparing whole Times New Roman Family to the Times New Roman Regular. If you take a close look you will notice that the Unicode ranges/blocks differ (TNR Regular does not provide Arabic, Hebrew, Cyrilic, etc. characters).
There exist a few different TNR variants, since already Microsoft's naming "Times New Roman" is in reality TrueType version of Times New Roman PS, a narrower variant of Monotype's classic Times New Roman typeface.
However to answer your question directly - the two font's you compare have the same glyphs, just that you compare the whole family to a single subset.