Is the God of the Old Testament and the New Testament the same God? Or are they different?
Firstly there is only one God with many representations...
My understanding is Yah - Avah means 'Lord To Be' (physically), Yehoshua means 'Lord that Saves'; these are both representations from the One God Most High (El Elyon).
Though Yeshua's actions, and statements (Matthew, Mark, Luke) fulfills prophecy precisely in a Hebraic context; Christianity (John, Paul, Simon peter) is a malformed mixture of Pharisaic and Hellenistic ideas, which then does have a different god being described.
As we dissect any religious texts with many different authors, we can always show many different theologies, as each author has their own slightly different slant on things as well.