What's the difference between these most basic, first-order models for estimating runoff?
The models are similar in that they simplify the rainfall-runoff relationship into a few parameters, and essentially use some kind of first-order coefficient to partition rainfall into infiltration and runoff. The differences are mostly in what other parameters are brought in, such as initial abstraction in the SCS method (i.e. some capture of antecedent moisture conditions). You may wish to pick the method that best captures the information you wish to include in the calculation (or have available), such as drainage area. You can find more details about the implicit assumptiopns and ranges of applicability for each method (e.g. rational method).
In terms of other methods, there are many methods that use some similar forms of soil moisture, soil saturation, etc. to estimate the rainfall-runoff partition. For example, take a look at the Infiltration section (4.2) of the Raven Hydrologic Modelling Framework manual v2.8.
As to whether they are empirical, likely depends who you ask. Empiricism is likely more of a spectrum than a binary system. These are not simply linear regression equations and do capture some physics, although they are still clearly at least partially empirical since the coefficients are basically based on experiments to best fit results. I would likely describe them as empirical or semi-empirical, although I wouldn't spend too much time trying to split hairs for this question.