Does mass really exist?

says In a nutshell that the mass actually never existed. We are actually measuring energy all along

If energy exists then since $E=mc^2$ mass must exist also. Something that exists cannot be equal to something that does not exist.

Why do I need the concept of mass now?

Indeed, the concept of relativistic mass is unnecessary as it is just another name for the total energy. However, in modern nomenclature "mass" refers to the invariant mass which is a distinct concept given by $m^2 c^2 = E^2/c^2 - p^2$. For $p=0$ this equation reduces to the famous one, but it is a more general equation. This concept is the concept used when particle physicists talk about the mass of a particle, and it is physically and conceptually distinct from energy.

Why do I need the concept of mass now?

The mathematics of special relativity is based on four vector algebra, the four vectors being defined as $(E, p_x,p_y,p_z)$. The length of this four-vector is invariant under Lorentz transformations, and is called the invariant mass of the particle/object.

$$sqrt{Pcdot P}=sqrt{E^2-(pc)^2}=m_0c^2$$

For velocities much smaller than the velocity of light, this is identical with the classical definition of mass.

The $E=mc^2$ mass is confusing and not useful in studying particle interactions, as it depends on velocity, so is not Lorentz invariant. It is useful for calculation of fuel needed in future space journeys and such speculations.

Mass is the proportional coefficient between force and acceleration. In everyday life it's necessary,but in models without particle-like things you can throw it away. In some particle models it is interchangeable with energy, but the buildup of the model needs its concept.

In everyday life, mass is the property of an object that causes it to resist acceleration. I've heard that results from the way that matter interacts with the Higgs field. Mass also plays a role in the effect we refer to as gravity. We may not understand what mass is, but its effects can be observed and measured. The same can be said for pretty much any entity in physics.

Mass is the energy in the rest frame of a system divided by c$^2$. In order to be useful, mass should be constant at the time and energy scale of your experiment. For example, a football has mass. If you kick it it heat up a little, but this change in mass can be neglected when you play.

When you look inside a massive object you will find that not only the mass of the constituent parts contributes but also the binding energy of these. For example the mass of a hydrogen atom consists is m$_p$ +m$_e$ - E$_b$, where E$_b$, positive by definition, contains electro-nuclear potential energy, electron kinetic energy and a proton kinetic energy.

The $m$ in the equation $E=mc^2$ is the rest mass and should be written as $m_0$. $E$ is also equal to $hnu$ where $nu$ is the frequency. So $m_0c^2 = hnu$. This implies that the rest mass $m_0 = frac{h}{c^2}nu$. The mass of an object at rest is proportional to frequency. We can think of mass as a vibration, even at rest.

The point is not that mass doesn't exist, but that matter is not something fundamentally different from energy.

Energy can take many forms, and one of those happens to be when it is concentrated into a tiny space.

Imagine light ("pure energy") bouncing back and forth between two mirrors in a box. Only you know what is inside the box; to anyone else it is a mystery.

If we continually increase the amount of light in the box, the energy continually increases, and the mass of the energy in this arrangement can be made quite large.

Now imagine the energy is increased so much that whatever mass the box and mirrors had becomes negligible.

Or equivalently, imagine that the bouncing light somehow becomes a self-contained process, and the box and mirrors are no longer needed. (In fact they weren't, they were simply to help you imagine what is happening.)

You now have what looks like an object with substantial mass. No one can tell that the mass is simply results from the energy contained in what for convenience we call a particle. But you know the truth of the matter.

Energy comes in various forms. For example, there is the kinetic energy of a particle in motion. There is the field energy of the electric field in a capacitor, or the magnetic field in a solenoid. Also there is the energy carried by a particle even when it is not moving. That energy we call mass. Or, to be more precise, the rest mass of a particle is equal to the total energy of the particle when it is not moving, divided by $c^2$. Another way to define rest mass $m$ for a particle is via $$ E^2 - p^2 c^2 = m^2 c^4 $$ where $p$ is the momentum and $E$ is the total energy, and now the particle may be in motion. Your question is phrased "does mass exist?" Well of course it exists; I have just defined it and it has many easily observed properties. But I understand that what you meant is "given that we can always invoke its close relationship to energy, can we imagine doing all of physics without invoking the concept of mass?" The answer to that is "in many respects yes, but ultimately no". The "in many respects yes" is because in many situations you can indeed just replace rest mass $m$ by $E_0 / c^2$ (where $E_0$ is rest energy) and all the formulae and results are correct. But in quantum field theory these two concepts enter differently. Each type of quantum field is found to be associated with a certain amount of rest mass associated with the excitations of the field. And this quantity in turn is related to the coupling between the field and another one called the Higgs field. To track all these behaviours without mentioning mass would be, I suppose, possible, but artificial. It would be like refusing to keep clear in the physical description the fact that the rest mass plays a special role in the physics of the quantum field overall; it is not just kinetic energy by another name. It is still a form of energy, but if one does not name it by the label "mass" then one is losing sight of a relevant distinction. It would be like refusing to call elephants and mice by different names because they are both mammals. They are indeed both mammals but that does not mean they are exactly alike in all respects.

Having said that, when we come to composite systems then the kinetic energy of the internal parts does contribute to the rest energy of the whole system, so here the distinction between mass and energy is less strong. For example the rest mass of a proton is almost entirely owing to the kinetic energy of its constituent quarks and gluons. And the rest mass of the Sun includes a contribution from all the light propagating around inside it. In these sorts of examples you can indeed say that you do not need to invoke the concept of mass; just tracking the energy is fine.

The gravitational effects involve one more subtlety. In general relativity the gravitational effect includes a contribution from energy and a contribution from pressure and stress in the gravitating body. But for a given distribution of energy and stress it does not matter what form the energy is in.