Is the tower bridge the same as the London bridge?

Absolutely not! London Bridge is an ordinary modern (1973) bridge which spans the river Thames. Its predecessor, built in 1820, is now located at Lake Havasu in Arizona. Tower Bridge, completed in 1894, also spans the river Thames and is the only bridge across the Thames which can be raised to allow large ships to pass underneath it, although that doesn't happen very often now. The original Victorian mechanism can still be seen in working condition in the bridge's museum. The bridge is quite distinctive, with two tall towers at each end and is situated next to the Tower of London from where it got its name.
No. They are two separate and very different bridges. Foreigners often think that Tower Bridge is London Bridge, but they are very wrong.