One of the major differences between the Sermon of the Mount and the 10 Commandments is the way that the two were given. The 10 Commandments were given to Moses to spread to the people. The Sermon on the Mount was spoken word by Jesus, given to his disciples. Another difference was God was not seen when passing the commandments to the people, where as Jesus was seen by the people and performed miracles. With these two differences we see the changes that had taken place over the years since the first set of commandments had been given. .
The 10 commandments were seen as way to teach people the basic things that should not be done. With those commandments we see the 10 major things the Lord did not approve of. Those rules were given as a backbone, not to be fully explained but to be understood. If you broke any of those rules the punishment would be hell. Jesus brought the gift of salvation with his word (which was a gift from God). Under these circumstances the Sermon on the Mount had more insight into what was being spoken about then the 10 Commandments. .
The Sermon spoke of more subtle ways to receive your blessings, and how to join the Kingdom of Heaven. The Commandments were put into place because God knew his people would not follow the righteous path, so he made the decision to place the rules so the path to righteousness could be followed. Jesus taught the people how to maintain these rules now that the world had evolved and matured. With every line in the Sermon on the Mount you become deeper in tune with the word of Jesus and have a firm understanding of what it is he asks of his disciples as well as his followers.
Another major difference was the time that was spent. Moses, when receiving the Commandments was gone for a very long time while the Lord gave the Commandments and an explanation as to why Moses would not make it to see the new land. Jesus on the other hand did not leave the people when giving his commandments and healing.