There are two main reasons for the difference. The first reason is that the two names are slightly different in Hebrew. The Old Testament figure is יְהוֹשׁ֫וּעַ "Yehoshua", while the New Testament figure is יֵשׁ֫וּעַ "Yeshua". The two names come from the same root vocabulary. The New Testament version is considered a later form of the name (it appears in the later texts of the Hebrew Bible instead of the older version)

The second reason is because Joshua is primarily known to us from the Hebrew texts, and Jesus is primarily known to us from Greek texts, where the names are יְהוֹשׁ֫וּעַ "Yehoshua" and Ἰησοῦς "Iesous", respectively. As the Bible was translated into more common languages, the names became Anglicized based on the way they were written in their respective original texts, thus יְהוֹשׁ֫וּעַ "Yehoshua" becomes "Joshua" (hopefully you can see how the sound is similar) and the Greek Ἰησοῦς "Iesous" becomes "Jesus".

So basically it's mainly because the two names come to the English language through very different means.