Q:

Why doesn't $E = mc^2$ contradict the conservation of mass principle?

I was watching this very interesting video: The mathematics of weight loss
and at 12:35 the presenter says: "(...) people think that you can turn atoms into energy. It's one of the founding principles of modern chemistry: you cannot turn atoms into pure energy. It's called the conservation of mass."
Einstein's famous equation immediately came to my mind: $$E = mc^2.$$
Isn't that equation saying you can turn mass into energy and vice versa? Doesn't that contradict the conservation of mass principle then?

A:

It's true, it does violate the conservation of mass.
This conservation of mass principle was established before Einstein showed his famous equation of $E=mc^2$. So today we say that in a closed system mass-energy is conserved. That is the sum of mass and energy in a closed system is constant. 
( If you still like to think of the conservation of mass then think that mass and energy are equivalent and hence if you say that mass, which is equivalent to energy is conserved then the principle still holds).
And also it's not true that you can't turn atoms into pure energy, you absolutely can. You just collide them with equal mass of antimatter. As almost everything around is matter hence it makes sense that in practice it is very difficult to turn atoms into pure energy.

A:

There is no conservation of mass principle. Mass is approximately conserved in chemical reactions, the context of the remark.
Actually, a better way of stating the notion that the speaker is trying to get across in chemistry is to state it stoichiometrically: the number of atoms of each and every elemental species making up reactants and reaction products is unchanged by chemical reaction. If you have 20 hydrogen atoms and 10 oxygen atoms (making up ten hydrogen molecules and five oxygen molecules) before their reaction, and they combine to make water, then you still have 20 hydrogens and 10 oxygens after the reaction, only they are in 10 water molecules.
In the hundred years since science thought that mass was conserved, mass has become a less and less important notion in physics. Mass now has only one rigorous use in physics as the notion of rest mass:

The rest mass $m_0$ of a system equals the total energy of that system measured from a frame that is at rest relative to the system (in natural units - in SI units we have $E = m_0\,c^2$). "At rest" means that the system's total momentum is nought in this frame;
Newton's second law becomes $\mathbf{F} = \frac{\mathrm{d}\mathbf{P}}{\mathrm{d}\,t}= \frac{\mathrm{d}(m_0\,\mathbf{u})}{\mathrm{d}\,t}$, where $\mathbf{F}$ is the four force and $\mathbf{u}$ the four velocity.

But even rest mass is not conserved, and it is not linearly additive for composite systems. For example, two photons, each of energy $E$ moving in opposite directions relative to our frame, each have a rest mass of nought. But the system as a whole, from a frame where the two have equal momentums (which happens to be ours), has a rest mass of $2\,E/c^2$. 
For everyday systems undergoing chemical reactions, conservation and linear additivity for composite systems are both good approximate properties of the mass notion. 
Nuclear reactions change that idea altogether. Typically products of an exothermic nuclear reaction have a few percent mass deficit compared to the reactants. Even in chemical reactions there are differences between the masses of reactants and products,  but the differences are tiny.
Mass only seems to be constant, because if a system only changes its rest mass significantly by releasing / taking on quantities of energy that are far greater than we see in the everyday world.

