[[What I am about to say might be wrong, and doesn't cover the difference between virtual and on-shell particles which I don't understand very well. Please correct me if I make any mistakes...]]
What is important is to keep in mind the difference between the Higgs field and the Higgs boson.
You sometimes hear that the Higgs field permeates space. It is everywhere. It is between the spaces between electrons in atoms.
But, what is a field? A field is just a series of values, a complex number in the case of the Higgs Boson, tacked to each point in space.
On the other hand, what is a boson-- what is a particle? A particle is an excitation of a field. A particle, a boson, is a ripple in the values of the field.
The Higgs field permeates space. But it is flat. It takes the same value everywhere. The "empty space" in an atom does contain the Higgs field, but it's as far as I know it's all a flat ocean of Higgsness between the electrons.
The "Higgs Boson" we want to find is a ripple in the Higgs field. This ripple could be created in a particle accelerator collision. But the Higgs particle, the ripple, isn't something which is important in physics, I don't think, because it doesn't normally or commonly happen in nature. We want to find the Higgs particle so that we can measure various properties of the Higgs field. But it's the flat field which does all the important physics stuff.
