Is London bridge and tower bridge the same bridge?

No, they are completely different. London Bridge is famous because of the childrens' nursery rhyme 'London Bridge is falling down' but the present bridge is a very modern (1973) structure. Tower Bridge, a much more interesting bridge, was built in 1894 and is the only bridge in London capable of opening to allow taller ships to pass underneath. Many foreigners think that Tower Bridge is London Bridge.