Why is invariant mass preferred over conserved one? In Special Relativity, there are two kinds of mass one can define for a system, right?:
$$m_{rel} = \frac{E_{tot}}{c^2}$$
and
$$m_0 = \frac{\sqrt{E_{tot}^2 - p_{tot}^2c^2}}{c^2}$$
Both have their pros and cons. The first one is conserved (because $E_{tot}$ is) but not invariant. The second one is invariant (because $E_{tot}^2 - p_{tot}^2c^2$ is) but not conserved.
Usually, the second one is preferred over the first. Why is that? Why is in-variance preferred over conservation?
 A: The invariant mass $m_0$ is conserved by definition of its being a constant in time. There is no other equation for $m_0$ but $m_0=\text{const}$.
The mass $M_0$ of a compound system can be calculated (expressed) via the masses of interacting particles.
A: To an extent this is a matter of opinion since both invariant and relativistic mass can be used in calculations. However I suspect the opinion of most physicists these days would be that the invariant mass is a more useful concept.
If we start with Newtonian physics then we know that the basic equation of motion is Newton's second law:
$$ F = ma = \frac{dp}{dt} $$
If we now consider relativistic velocities then we find this no longer correctly describes the motion. However, if we use the relativistic mass $m_r$ we find the equations do work. This I suspect was why the concept caught on originally. If you regard Newton's laws as fundamental then it seems natural to redefine the mass to keep Newton's laws working.
But the force, acceleration, momentum, etc used in Newtonian mechanics are three-vectors and we know that special relativity is most naturally formulated using four-vectors. Indeed if you use four vectors you can simply write:
$$ \mathbf F = m\mathbf a = \frac{d\mathbf p}{d\tau} $$
and now $m$ is the invariant mass i.e. the same as the mass of the object in the objects rest frame. So if you regard four-vectors as fundamental then it seems natural to redefine Newton's laws and keep the mass constant.
As to which is better ultimately that is a matter of opinion, though I doubt you will find many physicists today who consider the concept of relativistic mass better. The invariant mass has a natural physical interpretation as the norm of the four-momentum (which immediately explains why it is an invariant since the norm of any four-vector is a scalar invariant).
Using this formalism also gives us the relativistic equation for the total energy:
$$ E^2 = p^2c^2 + m^2c^4 $$
that applies to everything - massless particles as well as massive ones. Unless you subscribe to this view you have a problem explaining why massless particles can have energy without any mass.
