These names end in "sheba" - a word that means oath. "Sheba" appears in many place names in the Bible, such as Beersheba, meaning "the well of the oath". It also appears in names such as Bathsheba, which means "daughter of an oath."
The difference in these names is in the way that they begin. "Eli" is the word for "God" while "Jeho" is taken from "Jehovah" and is the actual name of God. So, Elizabeth/Elisabeth/Elisheba mean "oath of God" while Jehosheba/Jehoshabeath mean "oath of Jehovah," using God's personal name. The difference between Elizabeth/Elisabeth and Elisheba is that the former is the Greek rendition of the latter [and apparently] original name.
In the Bible, there are three woman who bear versions of this name.
The first is Elisheba (אֱלִישֶׁבַע), wife of Aaron the priest. Elisheba was also the sister of Nahshon, who was the captain of Judah at the time of Moses. (Exodus 6:23 and Numbers 2:3) Exodus 6:23 says that Elisheba had four sons, Nadab and Abihu, Eleazar and Ithamar. The two eldest sons "fell dead before the LORD when they made an offering with unauthorized fire before him in the Desert of Sinai..." (Numbers 3:4). Elisheba's younger sons lived and served as priests with their father Aaron in the Tabernacle.
The second is Jehosheba (יְהוֹשֶׁבַע), wife of Jehoiada the priest. Jehosheba was also the daughter of King Jehoram of Judah. (2 Chronicles 22:11) Jehosheba rescued her nephew, Prince Joash, from death at the hands of his grandmother. The little prince was hidden in the temple until he was 7 years old and then he was crowned king. King Joash was said to have done what was right during the years that priest Jehoiada was alive; however, after Jehoiada's death, the king no longer did what pleased God. God sent prophets to warn King Joash, and even sent the son of priest Jehoiada to warn the king. Rather than listening, King Joash had Zechariah stoned to death in the temple. (2 Chronicles 24:2, 18-22)
The third is Elizabeth (Ἐλισάβετ), the wife of Zechariah the priest. Elizabeth was the mother of John the Baptist. Elizabeth's husband Zechariah was serving in the temple when the angel Gabriel came to deliver the news of John's birth; but Zechariah didn't believe that the news could be true because he and Elizabeth were old. Because of this, Zechariah was unable to speak until after the birth of John. People wondered at the marvelous circumstances of the birth and asked one another what John would grow up to be. (Luke 1:5-25, 57-80)
Abraham --> Isaac --> Jacob --> Judah --> Perez --> Hezron --> Ram --> *Amminadab* (father of Elisheba) --> Nahshon --> Salma --> Boaz --> Obed --> Jesse --> David --> Solomon --> Rehoboam --> Abijah --> Asa --> Jehoshaphat --> *Jehoram* (father of Jehosheba) --> Ahaziah --> Joash ...
(Hehe, lj spell-checker did not like spell-checking this post!)