In comparison to nanoplasmonic structures, resonant high-index dielectric nanoantennas hold several advantages that may benefit nanophotonic applications, including CMOS compatibility and low ohmic losses. One such application area might be label-free refractometric sensing, where changes in individual antenna resonance properties are used to quantify changes in the surrounding refractive index, for example, due to biomolecular binding. Here, we analyze and compare the sensing performance of silicon and gold nanodisks using a common and unbiased testing framework. We find that the all-dielectric system is fully capable of effectively monitoring small changes in bulk refractive index and biomolecular coverage, but the sensitivity is five to ten times lower than the plasmonic counterpart. However, this drawback is partly compensated for by a more linear response to adsorbate layer thickness changes and an approximately four times smaller susceptibility to photothermal heating. Finally, dielectric sensors may show promise if certain strategies are employed to improve their performance, which could thus bridge the gap between the two systems.