I've had this conversation with so many people now, and everyone seems to have their own definition and opinion. I don't believe there's an actual specific difference that you can pinpoint 100% of the time in our current skincare world. It used to be more simple, but our face oils now often double as, or replace, serums. Formulators and brand owners can really name their products whatever they like, with little regulation around specificity. I think of it in this very simple, broken down way (and not everyone will agree):
A face oil adds hydration and nutrition. A serum does something bonus - brightens, slows the onset of lines and wrinkles (The Youth Dew, via CoEnzymeQ10), tightens, exfoliates, etc. Serums can be essentially a face oil blend with an extra ingredient(s), or it can be an oil-free base of aloe or an emulsion of oils and floral waters or any number of other things. Serums vary wildly in texture and use. Some replace your face oil (because it's basically a super-power one), and others do not wear well alone at all and must be layered with an additional face oil or cream.