Except for imagery and rhyming couplets, there is more to contrast between these two poems than there is to compare. A few points to consider are these. In Browning's "My Last Duchess," he makes significant use of enjambment while Byron's "The Destruction of Sennacherib" employs stops at the end of each line. Enjambment is a technique that causes the thought begun in one line to carry over to the following line. This adds to a speedy pace of reading; to tension; to suspense; and to the complexity of the thought:
How such a glance came there; so not the first
Are you to turn and ask thus. Sir, 't was not
Her husband's presence only
Line-end stops may be commas, semicolons, colons, or periods. Lines with end stops give a more measured pace to the poem, thus creating emphasis, which can heighten tranquility ("When the blue wave rolls nightly on deep Galilee. / ... / Like the leaves of the forest when Summer is green,") or drama and intensity:
For the Angel of Death spread his wings on the blast,
And breathed in the face of the foe as he passed;
And the eyes of the sleepers waxed deadly and chill,
Browning uses rhyming couplets, with no stanza breaks, starting with aabbccdd and continuing through the sequence to the end couplet of "rareity" / "for me." Byron similarly uses rhyming couplets but with stanzas in quatrains with aabb ccdd through to the end couplet of "sword" / "Lord." Byron relies heavily upon Biblical and ancient allusion: Ashur, Angel of Death, Assyrian, etc., whereas Browning relies heavily upon imagery: "for never read / Strangers like you that pictured countenance," etc. Byron also relies heavily upon imagery: "sheen of their spears was like stars on the sea," etc.
Byron, recounts in third person point of view the legend of an ancient battle when the King of Assyria, Sennaccherib, came in force against the Hebrew city of Judah. The Biblical text of the chronicles of Second Kings, in the 18th and 19th chapters, tells that the Assyrians, whose gods were Ashur and Baal and whose holy city was also called Ashur, were conquered by Jehovah's might without any warfare:
Now in the fourteenth year of king Hezekiah did Sennacherib king of Assyria come up against all the fenced cities of Judah, and took them. 2 Kings 18:13
Therefore thus saith the LORD concerning the king of Assyria, He shall not come into this city, nor shoot an arrow there, nor come before it with shield, nor cast a bank against it. 2 Kings 19:32
Browning tells a narrative as a first person dramatic monologue about the strange life and sudden death of a Duke's previous (last) Duchess. The story is told on the verge of his meeting his next Duchess pending the finalization of the wedding agreements:
... ample warrant that no just pretence 
Of mine for dowry will be disallowed;
In this psychological poem, the Duke is speaking with his bride's father's representative ("I repeat, / The Count your master's known munificence") while showing him the portrait of his "last Duchess": "That's my last duchess painted on the wall, ...." The nature of the two poems is similar in the use of rhyming couplets and imagery, but different on subject matter, point of view, narrative style, and much poetic technique.