God is the same yesterday, today, and forever (Heb. 13:8 ). However, the way he deals with people changes depending on what the best solution is in a particular situation. You set the Old Testament against the New Testament, as though we are talking about the same human situation and it is God that has changed; the truth is exactly opposite that. In the Old Testament times we are dealing with the literal children of Abraham, Isaac, and Jacob. This was a specific people with specific positives and negatives. One of their positives, for example, was their ability to understand themselves as an ongoing community spanning many generations, and not just a collection of individuals who happened to be alive at that moment. One of their negatives, though, was a tendency to be led astray by wickedness in others. The people around them, as it so happened, were exceedingly wicked. Thus, God would sometimes tell the people he had chosen to work through to kill certain other groups off.
But that by no means is the end of it. On the contrary, God often times used other groups to conquer Israel, in an attempt to humble them. These were very rough times, day to day life (just trying to live another day) was hard, and wickedness prevailed. In ruthless times, God used ruthless measures; to do otherwise while still allowing free will would have been as effective as doing nothing at all. "There is a time to kill, and a time to heal" (Eccl. 3:3) and this was not the time for healing as happens after the coming of our Lord. The Son of God had not yet come, and so the people of those times (e.g., King David) saw even good things mostly in shadows and mysteries, and survived as best they could, with God condescending in such a way to make sure they survived.
But then the God-man came. His coming transformed and transfigured all of human nature, and every human being's ability to change. It's not that God changed between the Old Testament and New Testament, but that man was changed by God: the God-man changed the rules that man played by, not those that God played by. Consider what Jesus said about divorce:
"And he answered and said unto them, 'What did Moses command you?' And they said, 'Moses suffered to write a bill of divorcement, and to put here away.' And Jesus answered and said unto them, 'For the hardness of your heart he wrote you this precept. But from the beginning of creation God made them male and female... What therefore God hath joined together, let not man put asunder'" (Mark 10:3-6, 9)
Moses did not make things up as he went along, but God himself condescended to allow a certain violation of his will (cf St. John Chrysostom, Homily 17 on Matthew). But, again, things of the divine dispensation (not the divine essence) changed between the Old Testament and the New Testament.
God was still harsh at times in the New Testament, as certain passages indicate. The book of Revelation is probably one of the harshest books in the entire Bible, and Jesus was both physically violent (Jn. 2:15) and verbally condemning of entire cities (Matt. 11:20-24). God was the same in the New as he was in the Old Testament; but what he asked of us changed. "A new commandment I give unto you, that ye love one another" (John 13:34) This is not a new commandment in letter (Deut. 6:5; Lev. 19:18; etc.), but new because the energizing power of the Holy Spirit was to come, after Christ had completed the transformation of human (and cosmic) nature via his assumption, death, and resurrection.
The following is also recorded in the Scripture: "And some of the Pharisees which were with him heard these words, and said unto him, Are we blind also? Jesus said unto them, 'If ye were blind, ye should have no sin: but now ye say, We see; therefore your sin remaineth." (John 9:40-41) And, "If I had not come and spoken unto them, they had not had sin: but now they have no cloke for their sin. He that hateth me hateth my Father also. If I had not done among them the works which none other man did, they had not had sin: but now have they both seen and hated both me and my Father." (John 15:22-24)
These two passages articulate the fact that God deals with different peoples in different ways, depending on the circumstances. Sometimes the change in circumstances are of the people's own making, but sometimes God changes the circumstances. With the incarnation, death, and resurrection of Jesus, humanity was given the chance like never before to change their circumstances. Along these lines, St. Nicholas Cabasilas said:
...though men were triply seperated from God--by nature, by sin, and by death--yet the Savior made them to attain to Him perfectly and to be immediately united to Him by successively removing all obstacles. The first barrier He removed by partaking of manhood, the second by being put to death on the cross. As for the final barrier, the tyranny of death, He eliminated it completely from our nature by rising again. For this reason Paul says, "the last enemy to be destroyed is death" (1 Cor. 15:26). - The Life in Christ, Book 3
In a word, our earthly concerns and mortal enemies are no longer of any lasting consequence, and ours is now much more so a spiritual warfare (Eph. 6:10-18 ) Jesus said, "Think not that I am come to send peace on earth: I came not to send peace, but a sword" (Matt. 10:34), and yet this new sword was of a different type ( Heb. 4:12; Eph. 6:17; Rev. 1:16; 19:15)
