Why do we use the name "Jesus" if the apostles did not use it? The gospel says that God has given us one
name under heaven by which we a saved.
It appears throughout scripture that God
puts great importance upon names in both
old and new testament. The bible says
there is no other name by which we should
be saved, so if the apostles did not
attribute that to the name "Jesus" why
should we?

I understand that some say it does not
matter whether or not we say the right name
because God sees our intent but is that
just our own reasoning justifying our own
error?

The apostles spoke Hebrew, Greek and
Aramaic, they therefore either called him
Yeshua, Iesus or Yehoshua haMoshiach.

If the apostles did not caste out demons or
perform miracles in the name "Jesus" or
even use the name, why should we
continue to use it?
Your question seems to be referring to:


  And there is salvation in no one else, for there is no other name under heaven given among men by which we must be saved. - Acts 4:12 ESV


Which in context is referring to the name given in verse 10:


  let it be known to all of you and to all the people of Israel that by the name of Jesus Christ of Nazareth, whom you crucified, whom God raised from the dead—by him this man is standing before you well. (ESV, emphasis added)


Given this, your focus (while not actually explicit) can only be on the issue of whether this translation of the name of the Lord is correct. The context of verse 10 particularly the 'of Nazareth' component, reveals the Apostle Peter is denoting a person who is identifiable by that name and description.  The significance of the name is tied directly to the person it refers to.  Since 'all who call on the name of the Lord shall be saved' indeed from 'every tribe and tongue and nation', the appropriate name for them to use to call on him, must be the name used to identify that person in their language.