The Severi degree is the degree of the Severi variety parametrizing plane curves of degree d with δ nodes. Recently, Göttsche and Shende gave two refinements of Severi degrees, polynomials in a variable q, which are conjecturally equal, for large d. At q=1, one of the refinements, the relative Severi degree, specializes to the (non-relative) Severi degree. We give a combinatorial description of the refined Severi degrees, in terms of a q-analog count of Brugallé and Mikhalkin's floor diagrams. Our description implies that, for fixed δ, the refined Severi degrees are polynomials in d and q, for large d. As a consequence, we show that, for δ≤4 and all d, both refinements of Göttsche and Shende agree and equal our q-count of floor diagrams.