Every finite lattice is a Bounded lattice because for any Finite lattice, We can Prove that It has a least as well as a greatest element.
Every finite lattice is a Bounded lattice But the Converse is not true. i.e. A Bounded lattice may or may not be Finite. For example, $\left ( \left [ 0,1 \right ], \leq \right )$ is a Bounded lattice but Not finite. Here the least element is 0 and greatest element is 1.
Note that $\left ( \left ( 0,1 \right ), \leq \right )$ is Not Bounded lattice because there is No least or greatest element.