For instance, for nuclear fission we use E = mc^2. Uranium breaks into smaller atoms, but some mass has disappeared: it's because it has been radiated in the form of immaterial energy, hence nuclear plants and A-bombs.
On the other hand, m = E/c^2 gives an interesting interpretation to what inertial mass is. According to this equation, mass is actually "energy at rest". If you want to move your object, you have to give it some energy, so that the resulting energy of the body will result into the desired motion. It's kind of similar to how hot air (the energy you give) and cold air (system at rest) mix to form "medium-hot" air (system in motion).
And just because I like writing, note that we have no idea whether inertial mass (the m in m = E / c^2 and Newton's Third Law, F = ma) is equivalent to gravitational mass (the m in F_gravitation = G * m_earth * m / d^2), but experiments dismiss any difference bigger than 1 in 10^12. The principle of equivalency between these two masses is the fundamental postulate of general relativity.
In a nutshell, m = E/c^2 defines the inertial mass, and general relativity assumes it's equal to the gravitational mass.
Both explanations are valid, which doesn't mean they are contradictory.
It's kind of like one person in the train sees the person on the ground moving, even though from the person on the ground's point of view, it's the person on the train who is moving. Who's moving? Well, the interpretation depends on the referential you choose.
So here, you can say there is a referential "mass" and a referential "energy". They will give seemingly different interpretations, but which are actually equivalent.
(On a side note, there are indeed no differences between E=mc^2 and m=E/c^2, it's just that it's easier to conceptualize inertial mass as being energy at rest with the latter.)