The closest statement in the King James Version (KJV) to "Thus saith the Lord of hosts I will drive them away" is found in Jeremiah 24:8–10. This passage refers to God's judgment against the unrepentant people of Judah.
Jeremiah 24:8–10 describes the Lord's judgment on King Zedekiah, his princes, and the remaining people in Jerusalem and those in Egypt, stating, "surely thus saith the LORD". It details their removal to various kingdoms, becoming a reproach and a curse, and being afflicted by sword, famine, and pestilence until they are consumed from the land.
Jacob is the patriarch whose descendants formed the twelve tribes of Israel, while Judah is one of his sons and the tribe from which the Messiah and the kings of Israel descended. The key difference is that Jacob represents the nation of Israel as a whole, but it was Judah's lineage specifically that was chosen by God for kingship and, ultimately, for the arrival of Jesus Christ.
Jacob
Who he was: The patriarch, father of the twelve tribes of Israel.
His significance to the Lord: Jacob, through his descendants, received the covenant promises of God. He represents the entire nation of Israel.
Judah
Who he was: Jacob's fourth son and the ancestor of the tribe of Judah.
His significance to the Lord:
Royal Lineage: The "scepter shall not depart from Judah," signifying his descendants would hold the kingship.
Messianic Line: God chose Judah's line as the one through which the Savior, Jesus Christ, would be born.
"Lion of the Tribe of Judah": This title is given to Jesus in the New Testament, indicating his royal and conquering status as the Messiah.
The Key Distinction
Jacob is the father and the broader nation, while Judah is a specific tribe and the specific line chosen by God to carry the covenant promises related to kingship and the Messiah.
God made an explicit promise in Genesis 49:10 to Judah that he would be the ancestor of kings and the Messiah, a promise that has been fulfilled.