Rabbi Ahron Marcus in his book Kadmoniyot, pg. 131-142, discusses this subject at length. I'll translate some of the more relevant portions:
First of all, he strengthens the question by mentioning inscriptions discovered in Esarchaddon's palace. Esarchaddon took tablets from Tiglat-Pilesser III's palace to use for his own palace. Some of these tablets contained texts. These texts are now known as "the annals of Tiglath Pilesser". One of the texts reads:
"Mat Iudai...Azariau...[from there on he translates it into Hebrew:...twice and on the second Azaria and also Azariau and what remained whole is this and on the road of my journey I received the tax of the kings...Azariyahu the Judean and also...Azariyahu of the land of Judea...and these rebelled under the hand of Azariyahu..."1
That last part continues on to describe Azaryah leading a coalition of 19 kings against Assyria and defeating them (this story, which seems to have been a major event in the Middle East, is not mentioned anywhere in Tanach).
We therefore see that non-Jews also referred to Uzziah as Azaryah. So what's going?
"...and there are those that wanted to explain that Uzziah is a short-form of Azaryah that was used by the masses...and that is a mistake[n interpretation], because it was not the masses who called him so [Uzziah] but Yesha'ayhu the Navi himself, and also Zecharyah (14:5), and also among the prophets of the Kingdom of Yisrael we find Hoshe'a and Amos who call him Uzziah, we therefore have five witnesses against the Assyrian kings' texts...and the [bible] critics also missed the other names of Azaryah...we find the name Zecharyah which is how this king was called in Divrei Hayamim 2:26:5 when he would receive prophecies2. And they didn't understand that just as it is the custom of the Jews of Poland and Yemen today3 to use double names, one a holy name by which he [the man] is called to the Torah, and one name secular...[and then he brings evidence from across Tanach that other people had double names. For example, one of Shmuel's ancestors is named Uzziah in DH 1:6:9 but called Azaryah in DH 1:6:21; In Ezra 2:2 there's S'rayah while in Nechemyah 7:7 there's Azaryah; Yehoshafat in DH 2:21:2 had two sons with too-similar names, one named Azaryah and the other Azaryahu, therefore one must have had another name, and so forth]...and now we have found the reason that the Book of the Kings of Yisrael4 would use the name Azaryah and so too in the Assyrian kings' letters, for they didn't have but this name only, because it was the secular name that was known also to the nations, and also the meaning of the name "Uzziah" is more holy and exalted than Azaryah, but the name Zecharyah is the name given to him in the beit midrash of prophets where he was filled with the spirit of counsel and valor to defeat the great enemy the king of Assyria..."
The gist of what Rabbi Marcus wrote both here and later on, is that Azaryah was known by most people as Azaryah. It was his common/secular name, which is also how he was known by the Kingdom of Yisrael and by the Assyrians and other nations. However, he was known as Uzziah by the circle of prophets of his time, and it is based on their writings that he is called this in Divrei Hayamim, which is where his spiritual (and physical) rise and fall are described in detail (as mentioned above, not all of Azaryah's life is described in Tanach. Each book decided to write about different events for different theological reasons). Uzziah is a much more spiritual name (the strength of Hashem as opposed to the help/assistance of Hashem) which better reflects his character and his story of attempting to get close to Hashem (and doing so improperly) and his subsequent fall from spiritual greatness.
1 As with practically every Tanach-related archeological discovery, there are those that are skeptical of this interpretation of the text. For more on the text and some of the earlier opinions on it, see here.
2 Rabbi Marcus identified the one-mention prophet in the verse "He applied himself to the worship of God during the time of Zechariah, instructor in the visions of God..." (DH 2:26:5) as being the prophetic name of Uzziah/Azaryah, or in other words, the king had three names.
3 Late 19th century.
4 Earlier in the book he explains why he believes that Sefer Melachim was based mostly on the book mentioned in Tanach as Sefer Divrei Hayamim L'Malchei Yisrael (the Book of the Chronicles of the Kings of Yisrael) while Divrei Hayamim mostly based on Sefer Divrei Hayamim L'Malchei Yehudah (the Book of the Chronicles of the King of Judah).
