This is interesting because it's one of the few cases from antiquity where we have opposing accounts of an event. First we'll work through the Sennacherib Prism.The translation I'm using can be found at http://oracc.museum.upenn.edu/rinap/rinap3/corpus/, under Sennacherib 22.

On my third campaign, I marched to the land Ḫatti. Fear of my lordly brilliance overwhelmed Lulî, the king of the city Sidon, and (ii 40) he fled afar into the midst of the sea and disappeared. The awesome terror of the weapon of the god Aššur, my lord, overwhelmed the cities Great Sidon, Lesser Sidon, Bīt-Zitti, Ṣarepta, Maḫalliba, Ušû, Akzibu, (and) Acco, his fortified cities (and) fortresses, an area of pasture(s) (ii 45) and water-place(s), resources upon which he relied, and they bowed down at my feet.

The land Hatti refers to modern-day Syria and South Anatolia. This passage describes Sennacherib's subjugation of several areas of modern-day Syria, mostly in the region of modern-day Lebanon and the northern Levant. We can already see here that it's not randomly going out against Judah and Israel; he's on a broader project of asserting Assyrian control in Syria and the Levant. Until line 73 of this column, the annals talk about this campaign in places that don't involve Israel and Judah-mainly Ashkelon. Anyhow, we'll skip to that part:

(As for) the governors, the nobles, and the people of the city Ekron who had thrown Padî, their king who was bound by treaty and oaths (ii 75) to Assyria, into iron fetters and who had handed him over to Hezekiah of the land Judah in a hostile manner, they became frightened on account of the villainous acts they had committed. They formed a confederation with the kings of Egypt (and) the archers, chariots, (and) horses (ii 80) of the king of the land Meluḫḫa, forces without number, and they came to their aid.

This passage describes what Sennacherib's scribes would call a classic example of villainy on the part of the Assyrian's enemies-they attempted to destroy a ruler who was bound by a loyalty oath(adê) to Assyria. Having realized the depth of their villainy and their rebellion against their divinely appointed ruler, they become fearful and confederate with the king of Egypt.

In the plain of the city Eltekeh, they sharpened their weapons while drawing up in battleline before me. (iii 1) With the support of the god Aššur, my lord, I fought with them and defeated them. In the thick of battle, I captured alive the Egyptian charioteers and princes (lit. “the sons of the king”), together with the charioteers of the king of the land Meluḫḫa.

Naturally, of course, such villainy goes unrewarded and the rebels are defeated. Fairly standard trope in Assyrian royal inscriptions.

I surrounded, conquered, (and) plundered the cities Eltekeh (and) Tamnâ. I approached the city Ekron and I killed the governors (and) nobles who had committed crime(s) and (iii 10) hung their corpses on towers around the city; I counted the citizens who had committed the criminal acts as booty; (and) I commanded that the rest of them, (those) who were not guilty of crimes or wrongdoing, (to) whom no penalty was due, be allowed to go free.

The general thrust of this paragraph is the king's institution of royal justice; he rightfully(and I should make this clear that right now I am transcribing the arguments of the prism, not concurring with them) kills the men of the city who overthrew their rightfully appointed governor but pointedly permits the freemen of the city who had not involvement in this to go free.

)I brought out Padî, their king, from the city Jerusalem and placed (him) on the lordly throne over them, then I imposed upon him payment (in recognition) of my overlordship.

Again, the restoration of justice: The properly installed ruler, who Sennacherib had signed a treaty with, is now installed back on the throne.

Moreover, (as for) Hezekiah of the land Judah, who had not submitted to my yoke, I surrounded (and) conquered forty-six of his fortified cities, (iii 20) fortresses, and small(er) settlements in their environs, which were without number, by having ramps trodden down and battering rams brought up, the assault of foot soldiers, sapping, breaching, and siege engines. I brought out of them 200,150 people, young (and) old, male and female, (iii 25) horses, mules, donkeys, camels, oxen, and sheep and goats, which were without number, and I counted (them) as booty.

This paragraph recounts Sennacherib's action against Hezekiah. Sennacherib is acting not capriciously but because by accepting Padî as a captive Hezekiah had rendered himself insubmissive to Sennacherib and thus needed to be returned to his rightful submission to the universal king. The full might of Assyria is on display in this passage, with a massive complement of military forces, as is Sennachrib's ability to claim the tribute that is his due as king. Noteworthy here is that it is the very act of collecting tribute that defines Assyrian rule.

As for him (Hezekiah), I confined him inside the city Jerusalem, his royal city, like a bird in a cage. I set up blockades against him and (iii 30) made him dread exiting his city gate. I detached from his land the cities of his that I had plundered and I gave (them) to Mitinti, the king of the city Ashdod, Padî, the king of the city Ekron, and Ṣilli-Bēl, the king of the city Gaza, and (thereby) made his land smaller. (iii 35) To the former tribute, their annual giving, I added the payment (of) gifts (in recognition) of my overlordship and imposed (it) upon them (text: “him”).

This continues the description of Sennacherib's triumph over Hezekiah; he has given cities of Hezekiah over to his loyal vassals in reward for their loyalty.

As for him, Hezekiah, fear of my lordly brilliance overwhelmed him and, after my (departure), he had the auxiliary forces and his elite troops whom (iii 40) he had brought inside to strengthen the city Jerusalem, his royal city, and who had provided support, along with 30 talents of gold, 800 talents of silver, choice antimony, large blocks of ..., ivory beds, armchairs of ivory, elephant hide(s), elephant ivory, (iii 45) ebony, boxwood, every kind of valuable treasure, as well as his daughters, his palace women, male singers, (and) female singers brought into Nineveh, my capital city, and he sent a mounted messenger of his to me to deliver (this) payment and to do obeisance.

This paragraph describes Hezekiah's submission. Hezekiah is overwhelmed by Sennacherib's power(melammu beliya, "my lordly brilliance", with melammu having connotations of a sort of supernatural aura even if the king himself is mortal). Hezekiah brings his troops back into Jerusalem and submits as he ought to the yoke of Assur by sending a massive tribute of gold, silver, and other precious goods. At this point, there is no need to continue fighting or to record any continued fighting-the goal of the campaign is accomplished, with all rebel lands brought back into submission to Assur. In fact, this reads like a very typically standardized account of an Assyrian campaign-the king goes on campaign against rebels and oath-breakers, or those who do not return fugitives and by both his military prowess and god-granted radiance returns all rebels to their proper relation to Assur, as recounted in the first paragraph of the inscription:

Sennacherib, great king, strong king, king of the world, king of Assyria, king of the four quarters (of the world), capable shepherd, favorite of the great gods, guardian of truth (i 5) who loves justice, renders assistance, goes to the aid of the weak, (and) strives after good deeds, perfect man, virile warrior, foremost of all rulers, the bridle that controls the insubmissive, (and) the one who strikes enemies with lightning:

Looking at the biblical narratives in Kings and Chronicles, we note that many of the events described in the inscription are also found in the biblical accounts-the sack of cities, the tribute, the threats to Jerusalem(even if they take the form of speeches here rather than Sennacherib's lordly brilliance). The main difference, aside from the rather florid description of prayer to god and divine aid, is the plague. This difference tells us a lot about how the different accounts are framed. For Sennacherib, everything he set out to do was accomplished-by giving tribute(something even the bible records), Hezekiah demonstrated the proper submission to Assyrian rule. The bible, as a document seeking to emphasize God's divine role in securing Judah's place in history and in rewarding pious kings(and here too the documents mirror each other; both present their respective kings as pious and godfearing), attempts to showcase how God protected Hezekiah from Sennacherib by destroying Sennacherib's army. It doesn't really matter whether or not the plague happened(except from the standpoint of reconstructing military history), what matters is that the ultimate outcome was continued Assyrian hegemony in the levant and that the Bible's authors and the Assyrian annalists attempted to characterize that hegemony in different was according to their position in relation to it.