What's the difference between the ענקים and the רפאים of Tanakh? Both seem to be very large people. Is one a subset of the other?
Some Sources: Bamidbar 13:33 and Rashi there, Devarim 3:11
They were both tall people (giants), but they are a different group of people and they lived in different areas. One of the key verses is Deuteronomy 2:10-11 (texts from Sefaria):
הָאֵמִ֥ים לְפָנִ֖ים יָ֣שְׁבוּ בָ֑הּ עַ֣ם גָּד֥וֹל וְרַ֛ב וָרָ֖ם כָּעֲנָקִֽים׃ רְפָאִ֛ים יֵחָשְׁב֥וּ אַף־הֵ֖ם כָּעֲנָקִ֑ים וְהַמֹּ֣אָבִ֔ים יִקְרְא֥וּ לָהֶ֖ם אֵמִֽים׃
The Emim dwelt therein [in the land of Moab] aforetime, a people great, and many, and tall, as the Anakim; these also are accounted Rephaim, as the Anakim; but the Moabites call them Emim.
So the Repha'im are described to be tall as the Anakim, but they are neither the Ankim or a subset of them.
In your citation of Deuteronomy 3:11:
כִּ֣י רַק־ע֞וֹג מֶ֣לֶךְ הַבָּשָׁ֗ן נִשְׁאַר֮ מִיֶּ֣תֶר הָרְפָאִים֒ הִנֵּ֤ה עַרְשׂוֹ֙ עֶ֣רֶשׂ בַּרְזֶ֔ל הֲלֹ֣ה הִ֔וא בְּרַבַּ֖ת בְּנֵ֣י עַמּ֑וֹן תֵּ֧שַׁע אַמּ֣וֹת אָרְכָּ֗הּ וְאַרְבַּ֥ע אַמּ֛וֹת רָחְבָּ֖הּ בְּאַמַּת־אִֽישׁ׃
For only Og king of Bashan remained of the remnant of the Rephaim; behold, his bedstead was a bedstead of iron; is it not in Rabbah of the children of Ammon? nine cubits was the length thereof, and four cubits the breadth of it, after the cubit of a man.—
you point to the fact that Og was, in fact a Repha'ite, and, in fact Og was one of the most "famous" giants. His height is not described, here, but his bed;s length is described, thus, by inference, he was a giant similar to Anak. But, here, too, you can see that he lived in a different area than the Anakim. Also, refer to Genesis 14:5 that mentions that Chedarl'omer conquered the land of Repha'im, which explains this verse discussing that Og was the only survivor of that group.
As for Anak, see this Wikipedia article which will refer you to appropriate Torah verses.
They dwelt in the south of the land of Canaan, near Hebron (Gen. 23:2; Josh. 15:13). According to Genesis 14:5-6 they inhabited the region afterwards known as Edom and Moab in the days of Abraham. Their name may come from a Hebrew root meaning "strength" or "stature".
It seems that the Anakim get more mention than the Repha'im and they are the ones more known as the "giants" than Repha'im. This makes sense, as they were the first giants that the spies saw, and they also lived in Hebron, even at the time that Abraham travelled and lived there! We know that the Philistines were also giants. According to the Wikipedia article (I have to further verify its linked source), they were descendants of the Anakim. So, the Anakim were around for far more years than the Repha'im.
Rashi points out that these were different groups of of "giants". The Art Scroll commentary on Devarim 10-11 refers to Rashi and explained that the nation of Eimim were colloquially considered like Refaim "which was a different family of giants (Anakim)". However, they were not the Refaim whose territory Bnai Yisrael were going to conquer. The Eimim had been conquered by the children of Lot and their land was the inheritance of Moav. This was included as an explanation to ensure that Bnai Yisrael did not make the mistake of confusing the territories involved.