There is a substantial difference in my opinion. Laws rarely put forward a mechanism or seek a complete understanding of a relationship, while theories often do.
Ohm's law was a law long before the electron was discovered. It's an emperical linear relationship that just seemed to work. Since no mechanism is suggested, it's not possible to determine whether ohmic behavior is expected or not. In contrast, The Drude Model's explantion of ohmic behavior could be considered a theory. It makes an attempt to understand the underlying physics, and from this it is possible to predict whether a material will follow Ohm's Law or not.
I think modern scientists are a lot less likely to call any empirical relationship a law. Theory is still pretty widely used, and not for things that could be called laws.
You can say the "the theory of quantum chromodynamics," but not "the law of quantum chromodynamics". 'Law' implies a small number of simple equations, while theory allows for a much larger scope of complexity or rigor. The meaning is absolutely different. Modern science is pretty complex compared to what went on 200 years ago, so it's not surprising law has fallen out of favor.
"Laws" tend to be less rigorous and more narrowly applicable than "theories," though the colloquial meaning of the terms is switched.