Noether's relationship implies that for every conserved quantity, there is a symmetry, and vice versa.
In the real world, mass may be converted to energy. For example, uranium power plants convert about 0.1% of the uranium mass to a huge energy, according to the $E=mc^2$ formula. So up to the conventional factor $c^2$, the total energy - including the latent one - and the total mass is the same thing.
Its conservation is linked to the time-translational symmetry of the laws of physics.
In the old world "before Marie Curie", people didn't know any relativity or any other indications of relativistic physics such as radioactivity. So they believed that mass was conserved even if the energy is not included. In their understanding of the world, the total mass of the Universe was the sum of the rest masses of the electrons, protons, and other massive particles.
The conservation law for the mass defined in this way didn't lead to any symmetries because the definition only depends on parameters - masses of all point masses are parameters - and not on dynamical, time-dependence quantities such as the positions or velocities. That's why Noether's theorem didn't apply.
It is not a real flaw of Noether's theorem because, as we know today, the total mass defined in the old way - and neglecting the mass increases from velocity (kinetic energy) and other forms of energy - is actually not conserved. This is no coincidence; a world compatible with general relativity doesn't allow any mass-like quantities to be non-dynamical. All quantities describing particular objects have to be dynamical i.e. changeable, and that's why Noether's theorem always holds in the world.