Ether vs. Quantum Field Theory
A different point of view to Jamal's answer: I think what distinguishes quantum field theory, where each elementary particle in the particle table defines a field all over space time, from the luminiferous aether is Lorenz invariance.
The luminiferous aether theory was falsified by the Michelson Morley experiment because it was not Lorenz invariant.
In quantum field theory an electron traversing space time is described by a quantum mechanical wave packet (which means that what "waves" is the probability of existing at (x, y, z, t)), manifested by creation and annihilation operators acting on the electron field, and the expectation value defines the location of the electron as a function of (x, y, z, t). The same for a photon, riding on the photon field. The quantum fields though by construction are Lorenz invariant and thus cannot be identified with the luminiferous aether.
The aether or luminiferous aether was the medium postulated as that which light travelled through. For example, sound waves are mechanical waves, in that they travel through pressure and displacement of the medium, e.g. water or air.
Likewise, the aether was thought of as the 'material' which allowed the propagation of light. Now, if we look at the notion of a quantum field, it is understood as a field that propagates through all of space and time.
The physical interpretation of the quantum field itself may vary depending on what we are describing; in the case of electromagnetism, one quantises the 4-potential $A_\mu$. In the case of the Higgs, one quantises a scalar, $h$ (though it is more complicated than this in reality) which has dimensions $[h] = 1$, that is, of energy in natural units.
One can think of the excitations of the field $A_\mu$ as giving rise to quanta which we call photons, but in the general case of a quantum field, there is no notion of any wave propagating through it, and so it is a distinct notion from the aether.
In the case of the other quantum fields of the Standard Model, there is certainly no plausible relation whatsoever to the aether, as they don't have anything to do with electromagnetic radiation, but rather completely different particles.
In summary, just because two things are postulated to occupy all of space and time does not mean they are the same notion. Another example: a force field $\vec F$ is different from a scalar potential, $\phi$. In addition, the aether itself is a material and not a field; we could perhaps describe its distribution in terms of a mass distribution $\rho$ which is a field, but the aether itself is not a field.
Theoretical physics isn't really concerned with ideas very much; the research is really into building a theory, that is, a set of mathematical instruments which can be used to make accurate calculations and predictions.
The idea of aether may be reminiscent of the ideas of QFT. The fundamental difference is that the theory that came with aether actually made incorrect predictions, and so didn't turn out to be very useful. The mathematical tools of QFT, on the other hand, when turned into a specific theory such as QED, have stood the test of time in terms of accuracy.