So there are a good few things that go into this.

First off, the Greek translation isn't quite "Jesus". That's a mediaeval version of the Latinization of the Greek, which is properly Ἰησοῦς (Iesoûs). The first Latin versions of the name are some variant of Iēsūs. Both of these did not consistently or at all have the accent markings I'm using, of course. In contemporary documents, you'd more expect to see them in "majuscule" (capital letters, basically) as ΙΗΣΟΥΣ (Greek) and IESVS (Latin). The letter J has a messy history, and its use only began to be formalized after 1524. It's best evidenced from the mediaeval period, but there is some epigraphic evidence that suggests some differentiation from different letter Is in earlier Latin. (The letter I in pre-J Latin can represent, to use English sounds, an "i" sound or a "y" sound.)

It's also important to note that "Joshua" is, in its English pronunciation, a bit different to what His real name would have been. To the best of my knowledge, it'd be something a bit closer in sound to Yeshua; this is more how Hebrew and Aramaic are usually transcribed, and indeed Yeshua most likely spoke Aramaic. The "proper" Hebrew transcription is Yēšū́a, and the "proper" Aramaic is Yēšū́ʿ.

The reason this is important is that a lot of these sounds aren't in Greek. When Greek-speakers came into contact with Aramaic-speaking people and stories about people with Aramaic names, they had to write something. However, they couldn't represent it perfectly; the Ye- initial sound had to become the very-close-but-not-quite Ἰη-, and the middle -sh- (-š-) became a -σ-. This sounds similar enough because the Greek letter sigma (that one) is a "retracted s" sound, which sounds halfway between an English "s" and "sh". Try and move your tongue halfway between where it is when making either one of those sounds individually - that's pretty close. Ending on an -a sound would've been a bit weird for a man because of Greek grammar, and so we get the "Hellenized" -οῦς ending, which is nice and masculine.

As I said above, this then goes into Latin, and by the later 16th century, we in England appear to have begun to adopt the "Jesus" spelling over "Iesus", though the latter is still evidenced much later than that. The latest reference I can find in the OED is 1644. Our usage of the letter J changes to that "dj" sound we have now, and there you go. What's also important is that in the Greek, "Joshua" is also given as Ἰησοῦς. The name's quite rare, but it does turn up in the Septuagint (Greek Old Testament, if you don't know). There's no different form in the New Testament, at least in its canonical organization.

I still haven't got round to why one is now "Joshua" and one is "Jesus", you'll have noticed. All this background was important, though, because it makes the question even more puzzling. If not only the original Hebrew and Aramaic but also the Greek names are the same, why on earth are they different now?! I'm going to speak here pretty much entirely of English, but note that it varies across languages. In Modern Greek, the two still have the same name; this is also true of some Slavic languages.

The main difference appears to have come in the early modern period. If you look at a mediaeval manuscript, "Joshua" is still called by the form "Iesu Naue", which you might anglicize as "Jesus [son of] Naue". A good example is Bodleian Library MS. Laud Misc. 97, which is 9th c. and freely available if you want to look it up. By the 1611 publication of the King James Version, which had immense stylistic impact on later English-language Bibles, J- spellings still weren't universal, but a differentiation between "Iesus" and "Ioshua" is in place. This distinction wasn't hard and fast, though; some uses refer to Joshua (Naue) as "Iesus"; see the KJV's reference to Joshua while using the form Iesus in Acts VII:45!

Small discrepancies aside, the key factor appears to have been the weight of Humanist scholarship on the Hebrew OT combined with the length of tradition that used the form Iesus or Jesus. Note that both KJV usages of "Iesus" for Joshua are in the NT, which is written in Greek and would therefore have used Ἰησοῦς. Since early modern scholarship, and particularly Protestant Humanist scholarship on which the KJV was founded, had more of a focus on the Hebrew original of the OT over the Septuagint, it appears that they began to translate the "same" name differently based on their understandings of the phonologies of each language. Over time, this stuck; the sheer popularity of the Jesus form is considerable and goes back centuries. He's pretty much the most important guy in the NT, after all. Joshua is somewhat less important, and so changes in how scholars referred to him appear to have been less impactful.

As I said, I'm afraid I can't really talk on why this happened in other languages, though I suspect the reasons would be similar and based on the simple popularity of the Latin translation of the Greek (translation of the Aramaic!). In Spanish, for instance, it's Jesus and Josué, both of which also sound very different to the Hebrew and Aramaic originals and are different from each other.

Hope that helps!

Edit(s): cleaning up typos, bad wording, and so on.

Edit again: thank you very much to the people who awarded me! I think those are my first awards ever.

Here is the same question answered by u/kookingpot.