The Angel of the Lord is VERY different from the other angels. The Angel of the Lord is God Himself.

The Angel of the Lord speaks as God:

"And the angel of the LORD said unto her, I will multiply thy seed exceedingly, that it shall not be numbered for multitude."
~ Genesis 16:10

"And the angel of the LORD called unto him out of heaven, and said, Abraham, Abraham: and he said, Here am I. And he said, Lay not thine hand upon the lad, neither do thou any thing unto him: for now I know that thou fearest God, seeing thou hast not withheld thy son, thine only son from me . . . That in blessing I will bless thee, and in multiplying I will multiply thy seed as the stars of the heaven, and as the sand which is upon the sea shore; and thy seed shall possess the gate of his enemies; And in thy seed shall all the nations of the earth be blessed; because thou hast obeyed my voice."
~ Genesis 22:11 - 18

"And the ass saw the angel of the LORD standing in the way, and his sword drawn in his hand: and the ass turned aside out of the way, and went into the field: and Balaam smote the ass, to turn her into the way. But the angel of the LORD stood in a path of the vineyards, a wall being on this side, and a wall on that side. And when the ass saw the angel of the LORD, she thrust herself unto the wall, and crushed Balaam's foot against the wall: and he smote her again. And the angel of the LORD went further, and stood in a narrow place, where was no way to turn either to the right hand or to the left. And when the ass saw the angel of the LORD, she fell down under Balaam: and Balaam's anger was kindled, and he smote the ass with a staff. And the LORD opened the mouth of the ass, and she said unto Balaam, What have I done unto thee, that thou hast smitten me these three times? 9And Balaam said unto the ass, Because thou hast mocked me: I would there were a sword in mine hand, for now would I kill thee. And the ass said unto Balaam, Am not I thine ass, upon which thou hast ridden ever since I was thine unto this day? was I ever wont to do so unto thee? and he said, Nay. Then the LORD opened the eyes of Balaam, and he saw the angel of the LORD standing in the way, and his sword drawn in his hand: and he bowed down his head, and fell flat on his face. And the angel of the LORD said unto him, Wherefore hast thou smitten thine ass these three times? behold, I went out to withstand thee, because thy way is perverse before me:"
~ Numbers 22:23- 32

The Angel of the Lord identifies Himself with God and exercises the responsibilities of God:

And the angel of the LORD found her by a fountain of water in the wilderness, by the fountain in the way to Shur. And he said, Hagar, Sarai's maid, whence camest thou? and whither wilt thou go? And she said, I flee from the face of my mistress Sarai. And the angel of the LORD said unto her, Return to thy mistress, and submit thyself under her hands. And the angel of the LORD said unto her, I will multiply thy seed exceedingly, that it shall not be numbered for multitude. And the angel of the LORD said unto her, Behold, thou art with child and shalt bear a son, and shalt call his name Ishmael; because the LORD hath heard thy affliction. And he will be a wild man; his hand will be against every man, and every man's hand against him; and he shall dwell in the presence of all his brethren.
~ Genesis 16:7-12

And the angel of the LORD appeared unto him in a flame of fire out of the midst of a bush: and he looked, and, behold, the bush burned with fire, and the bush was not consumed.
~ Exodus 3:2

And an angel of the LORD came up from Gilgal to Bochim, and said, I made you to go up out of Egypt, and have brought you unto the land which I sware unto your fathers; and I said, I will never break my covenant with you. And ye shall make no league with the inhabitants of this land; ye shall throw down their altars: but ye have not obeyed my voice: why have ye done this? Wherefore I also said, I will not drive them out from before you; but they shall be as thorns in your sides, and their gods shall be a snare unto you. And it came to pass, when the angel of the LORD spake these words unto all the children of Israel, that the people lifted up their voice, and wept.
~ Judges 2:1-4

And there came an angel of the LORD, and sat under an oak which was in Ophrah, that pertained unto Joash the Abiezrite: and his son Gideon threshed wheat by the winepress, to hide it from the Midianites. And the angel of the LORD appeared unto him, and said unto him, The LORD is with thee, thou mighty man of valour. And Gideon said unto him, Oh my Lord, if the LORD be with us, why then is all this befallen us? and where be all his miracles which our fathers told us of, saying, Did not the LORD bring us up from Egypt? but now the LORD hath forsaken us, and delivered us into the hands of the Midianites. And the LORD looked upon him, and said, Go in this thy might, and thou shalt save Israel from the hand of the Midianites: have not I sent thee? And he said unto him, Oh my Lord, wherewith shall I save Israel? behold, my family is poor in Manasseh, and I am the least in my father's house. And the LORD said unto him, Surely I will be with thee, and thou shalt smite the Midianites as one man. And he said unto him, If now I have found grace in thy sight, then shew me a sign that thou talkest with me. Depart not hence, I pray thee, until I come unto thee, and bring forth my present, and set it before thee. And he said, I will tarry until thou come again. And Gideon went in, and made ready a kid, and unleavened cakes of an ephah of flour: the flesh he put in a basket, and he put the broth in a pot, and brought it out unto him under the oak, and presented it. And the angel of God said unto him, Take the flesh and the unleavened cakes, and lay them upon this rock, and pour out the broth. And he did so. Then the angel of the LORD put forth the end of the staff that was in his hand, and touched the flesh and the unleavened cakes; and there rose up fire out of the rock, and consumed the flesh and the unleavened cakes. Then the angel of the LORD departed out of his sight. And when Gideon perceived that he was an angel of the LORD, Gideon said, Alas, O LORD God! for because I have seen an angel of the LORD face to face. And the LORD said unto him, Peace be unto thee; fear not: thou shalt not die. Then Gideon built an altar there unto the LORD, and called it Jehovahshalom: unto this day it is yet in Ophrah of the Abiezrites.
~ Judges 6:11-24

In that day shall the LORD defend the inhabitants of Jerusalem; and he that is feeble among them at that day shall be as David; and the house of David shall be as God, as the angel of the LORD before them.
~ Zechariah 12:8

The Angel of the Lord accepts worship. Under no circumstances is man to worship anyone or anything other than the true and living God (Exodus 20:3-5; Matthew 4:10) God's angels reject worship. However, the Angel of the Lord accepts it, indicating He is God.

And he said, Nay; but as captain of the host of the LORD am I now come. And Joshua fell on his face to the earth, and did worship, and said unto him, What saith my Lord unto his servant? And the captain of the LORD's host said unto Joshua, Loose thy shoe from off thy foot; for the place whereon thou standest is holy. And Joshua did so.
Joshua 5:14-15

In several of these appearances, those who saw the Angel of the Lord feared for their lives because they had "seen the Lord." Therefore, it is clear that the Angel of the Lord is a theophany, an appearance of God in physical form.

The appearances of the Angel of the Lord cease after the incarnation of Christ.“The Angel of the Lord” is never mentioned in the New Testament. The appearances of the Angel of the Lord were manifestations of Jesus before His incarnation.

Jesus declared Himself to be existent “before Abraham,” (John 8:58) Jesus was, is, and forever will be the Living Word of God. Prior to God's Word appearing in print, the definitive Word, who later becams flesh (Jesus) was active and manifested Himself in the world as the Angel of The Lord.