What changed between the Old and New Testaments? In the Old Testament God meted out his justice through people, or he himself meted out justice immediately through his own hands:
All Scripture is quoted from the King James translation.

Exodus 19:12 and 13  And thou shalt set bounds unto the people round about, saying, Take heed to yourselves, that ye go not up into the mount, or touch the border of it: whosoever toucheth the mount shall be surely put to death:  13  There shall not an hand touch it, but he shall surely be stoned, or shot through; whether it be beast or man, it shall not live: when the trumpet soundeth long, they shall come up to the mount.
Exodus 2:12  And he looked this way and that way, and when he saw that there was no man, he slew the Egyptian, and hid him in the sand.
Numbers 14:11 and 12  And the LORD said unto Moses, How long will this people provoke me? and how long will it be ere they believe me, for all the signs which I have shewed among them?  12  I will smite them with the pestilence, and disinherit them, and will make of thee a greater nation and mightier than they.

However, with the advent of Jesus another precept was initiated.

Matthew 5:21 and 22  Ye have heard that it was said by them of old time, Thou shalt not kill; and whosoever shall kill shall be in danger of the judgment: 22  But I say unto you, That whosoever is angry with his brother without a cause shall be in danger of the judgment: and whosoever shall say to his brother, Raca, shall be in danger of the council: but whosoever shall say, Thou fool, shall be in danger of hell fire.

Here we find Jesus not only directing us not to kill, not to even berate them for what they say.
As a Southern Baptist I must wonder why the Same God, would change his Justice imposition so drastically.
Is there any explanation for this in any doctrinal belief in any denomination which addresses this?
Some quick, important notes about your references before answering your question directly:


Moses is not clearly justified in killing the Egyptian. He fled as a guilty man because he committed murder. There are many instance of men taking justice into their own hands, but this does not mean it was good for them to do so.
God continues to bring judgment and punishment under the New Testament. Remember Ananias and Sapphira (Acts 5) , and Herod (Acts 12) who were killed where they stood by God for their sin.


Perhaps these two notes together answer your question sufficiently and show that God is unchanging. He alone is judge, and he alone can delegate authority to judge. When comparing the Old and the New testament, however, we mustn't forget how the Cross is now a central feature of God's judgement. When Christ suffered the wrath of God, he was taking on wrath for sin yet to come as well as sin that had already passed. We know this because it was needful for him to die once. And I think in this is the answer to your question:


  ...but now once in the end of the world hath he appeared to put away sin by the sacrifice of himself.  And as it is appointed unto men once to die, but after this the judgment: So Christ was once offered to bear the sins of many; and unto them that look for him shall he appear the second time without sin unto salvation.  Heb 9:26-28 KJV


"It is appointed for men once to die, but after this the judgement. With the Cross, God has changed the timing of his judgement. Christ suffered the wrath of God, and now more than ever God is long-suffering towards sin until the end, so that many will be saved. Even throughout the Old Testament, God suffers long toward sin, waiting centuries sometimes before people have to answer for their rebellion. We as Christ's disciples are to imitate Him, not bringing condemnation before the appointed time. Kiss the Son, lest he be angry, and you perish in the way. God has not changed the terms of judgement, just the timing and frequency with the Cross.

Perhaps it would be well to ask another question regarding judgment in the New Testament (like that of Herod in Acts 12) and how it relates to God's plan for judgement at the end. That would be a very related, albeit separate, discussion.