Each equation uses a different definition of mass. e=mc^2 uses m="relativistic mass," which increases the faster the mass is moving relative to the observer. The second equation uses m="rest mass." The first equation was Einstein's, the second is used more these days because it is usually more convenient and less confusing.

Each equation is correct, given its definition of m.

If one starts from E = mc[itex]^{2}[/itex] and replaces m by [itex]\frac{m_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}}[/itex] where m[itex]_{o}[/itex] is the mass at rest, one gets the other version of the equivalence equation.