Interesting differences here.

The first one I notice is by saying “But seek ye first the kingdom of God,” the KJV implies the disciples of Christ do not have the kingdom of God and have not attained to God’s righteousness, since they are told to seek those things. 

The JST, on the other hand, by saying “seek ye first to build up the kingdom of God” assumes the disciples do have both the kingdom of God and God’s righteousness.  This states that since they have those things, their mission is to build up that kingdom and establish the righteous principles they have learned among its citizens and among as much of the world as they can.

Another difference I notice is the KJV makes it seem like the disciples are allowed to seek the things of the world as long as they seek the kingdom of God first.  But the JST, with the addition of “Wherefore, seek not the things of this world” clarifies that the disciples are not to seek the things of the world at all. 

So, the JST clarifies what the disciples have, what their mission is, and what they are not to worry about.