Do Not Kill
The law states "though shall not kill" (physical death) and will bring forth judgement
Do Not Commit Adultery
Refers to the actual act of adultery
Do Not Swear
Deuteronomy 6:13 / Leviticus 5:4
The law refers to a false oath/pledge
Exodus 21:22-25 / Leviticus 24:19-20
The law states "an eye for an eye" and "a tooth for a tooth"
Treatment of Neighbors/Enemies
The law states to love they neighbor and the religious leaders may have added "hate the enemy" for their own purposes.
Jesus says unjust anger with your brother will bring forth judgement
Jesus says that just looking upon someone with lust commits adultery in their heart
Jesus says just don't do it and there won't be a chance of it being false
Jesus says to turn the other cheek, to go the extra mile and to forgive and forget
Jesus says to love your enemy and to pray for those that use you
If you're accused of being radical because you're a Christian, that's fine. According to what Jesus says here in Matthew 5, that's exactly what it takes...being radical.