Actually we haven’t observed a difference between the speed of neutrinos and light, which sets an upper limit on their mass based on the precision of those measurements. The evidence of mass is much weirder, in the observation of neutrino flavor oscillation.

Neutrinos come in 3 flavors: electron, tau, and muon. We’ve observed that the flavor oscillates as a function of time, or equivalently as a function of distance from the neutrino source. The data is quite precise, and is perfectly explained by there being 3 neutrino mass states that are distinct from the flavor state. So a neutrino with a well defined flavor will have a superposition of the 3 mass states, with each flavor corresponding to a different admixture of mass states.

The flavor oscillations allow us to measure the difference in the mass values, but not the absolute masses. Technically it’s possible that the “lightest” neutrino is massless, and the other 2 mass states are nonzero. Without an absolute value for the masses we can’t rule this out.