I've been looking through the Septuagint and NT Greek manuscripts online and seen these different forms of the name for both Joshua and Jesus. Can someone please explain to me the differences?
This is really a rather elemental question about language, there is nothing special about this name. I would caution you not to make too much of what you find in the Greek manuscripts without actually learning Greek. Learning another language is more than just substituting vocabulary words. There are grammar patterns, ranges of semantic meaning, idioms, and even different logic that you will miss just looking at words. Worse yet, you won't know what you're missing and might easily come to totally false results. You are far more likely to come to wrong conclusions by applying your English language assumptions to a language that does not play by the same rules than if you'd just used a translation in the language you know.
Greek words use different endings depending on how they are used in a sentence. This is just the way the grammar works, indicating person, tense, voice, mood, case, number, gender, or degree. In this case Ἰησοῦς is the lexical form (or root word) and all the other forms are contextual variations based on it's use in a sentence.