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But mass is everywhere, so aren't there Higgs bosons everywhere too?

If so, what is the difference between detecting the Higgs boson via the presence of mass (i.e., "These particles have mass so there must be one or more Higgs boson nearby) versus detecting the Higgs boson as a separate particle?

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The excitation of the higgs field corresponding to higgs boson's like we are looking for at LHC are 'manifested' so to speak. The higgs that cause mass are virtual carrier particles that exist in a state of quantum in betweenness. Photons, for example are the carriers of the electromagnetic force. They are not the same, however, as the photons transmitted to your eye. So you don't see photons being traded between magnets when you hold them together, but can see the photons bouncing off the magnets from the sun. In the same way we can't see the higgs carrier particles, but can see (hopefully) the higgs bosons at LHC.

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