In the Latin version of the Gloria, the name Jesus is rendered as "Domine Fili unigenite Jesu Christe".
However, in the Latin Vulgate, the name of Jesus is rendered as "Iesus". What is the name of Jesus Christ in Latin? Why the difference?
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The phrase you quote has words in the vocative case.
Domine Fili unigenite Jesu Christe
The vocative case is used for address. That is,
O Lord, only begotten son, Jesus Christ
The particle O underscores this fact, that the phrase is in the form of address.
On the other hand, Iesus is in the nominative case. The nominative case is used for the subject of a sentence. So when Jesus is the subject of a sentence, the Latin word Iesus (or Jesus) will be used. An example of this would be,
Cum ergo natus esset Jesus in Bethlehem Juda in diebus Herodis regis, ecce magi ab oriente venerunt Jerosolymam,
When Jesus therefore was born in Bethlehem of Juda, in the days of king Herod, behold, there came wise men from the east to Jerusalem.
Here, the nominative form is used, because Jesus is the subject of the clause. When Jesus therefore was born in Bethlehem... I should mention that Jesus and Iesus are the same Latin word, it just depends on the edition of the Vulgate. Over time, the Latin I was sometimes replaced with J. So when you see Iesus or Jesus in Latin writing, know that it is unquestionably the nominative case. But when you see Iesu, it can either be vocative, dative, genitive, or ablative. Context helps you decide. In this case, Domine is unquestionably vocative, so we know that Iesu is as well.
Ancient Latin had no separate letter for the the vowel I and the consonant Y (J in German). They were both written as I. In Medieval Latin, though, a development took place that differentiated between the vowel I (written as I, i) and the consonant I (now written as J, j).
There is no difference, then, between Iesus and Jesus in Latin.
Jesu/Iesu, though, is different. Latin words had different endings depending on how they were used in a sentence. You can see from this page all the different endings for Jesus. Depending on how you use it, Jesu can be a genitive (of Jesus, Jesus'), dative (to/for Jesus), ablative (with/by/in Jesus), or vocative, (Jesus!).
The other answers are great, but let me try to show it in a slightly different way. In Latin, the word Jesus or Iesus (more on that later) has several different forms, as shown in a declension table like this one, reproduced below:
Case Form nominative Iēsus genitive Iēsū dative Iēsū accusative Iēsum ablative Iēsū vocative Iēsū
These forms – Iēsus, Iēsū, Iēsum – are different ways of saying "Jesus," according to their grammatical function in a sentence. In English we do this to pronouns (like the difference between he, his, and him), but in Latin it's done to regular nouns too.
The difference in the first letter, on the other hand, is a separate matter. The Classical Latin alphabet had no J. The letter I served as both a vowel and a consonant. As time went on, J was introduced and a distinction arose between I and J, with I representing the vowel and J representing the consonant.
Thus for practical purposes Jesus and Iesus are completely equivalent in Latin. And in English, where we'd say "Jesus," depending on the function of the word in the sentence, Latin authors would write Jesus, Jesu, or Jesum (or, alternatively, Iesus, Iesu, or Iesum).