Time and again I read/hear popular-level descriptions of the Higgs mechanism in which it is proclaimed that the Higgs field is "like molasses", offering resistance to particles moving through it. This is an awful analogy, and makes me cringe every time. Even non-physicists should immediately see why: a particle moving through molasses feels a drag force which will ultimately bring it to rest with respect to the molasses (in the absence of some persistent driving force). But the Higgs field fills all of spacetime, and thanks to Galileo and Newton, we know that in empty space, in the absence of forces, particles move with an arbitrary constant velocity (up to the speed limit imposed by Einstein, of course!).
The big difference is that the background value of the Higgs field is Lorentz-invariant — it doesn't define any absolute standard of rest.