The major difference between Metternich and Bismarck had to do with their attitudes towards nationalism. Bismarck embraced nationalism while Metternich saw it as a great danger.
In the time before the French Revolution, people had not generally seen themselves as part of a particular nation such as Germany or France. Instead, they identified with their own localities and with their rulers. A person in some part of Italy, for example, might be ruled by the sovereign of France but would not feel that this was wrong. They would not have felt that they should have been ruled by an Italian. Metternich wanted to preserve this system. Metternich was a prince (a title that he was given for his service, not one to which he was born) of the Austrian Empire. This was an empire that ruled many people of different nationalities. If the various people of this empire had come to feel nationalism, it would have made it very difficult for the empire to stay together. Metternich wanted to avoid this. He wanted things to return to the way they had been before the French Revolution.
By contrast, Bismarck wanted to encourage nationalism. Before Bismarck, Germany was not a unified country. Instead, it was split up into many little independent states. Bismarck felt that all ethnic Germans should be part of one country. He wanted to awaken their feelings of nationalism so that he could unify them. This would allow Germans to be ruled only by other Germans and it would also make Germany a much more powerful force in European politics.
Thus, Bismarck and Metternich had very different views on nationalism.