From the circuit level, the OD-3 and ODR-1 are completely different.
The ODR-1 is like a much more complex version of an OD that uses op amps and clipping diodes for everything. It has both hard and soft clipping, and way more EQ & filtering stuff happening than something like a TS does. The tone circuit is also a bit more sophisticated than either a TS or just the passive type that you'll find in many OD pedals.
The OD-3 only uses op amps for its hardwired bass/mid/treble EQ characteristics, and that all falls after the pair of clipping stages. The clipping stages themselves use jFETs, with the primary stage having them arranged as a 'differential'. The tone circuit in the OD-3 is just a simple passive affair that rolls off treble, and it falls after all of the op amp EQ shaping stuff (at the tail end of the circuit).
The OD-3 also differs from many OD designs in that it has what is a pre-gain notch filter of sorts, which attenuates mids at a corner freq of ~400Hz (IIRC, could be wrong about the specific freq).
...As cool as the ODR-1 is, I personally find myself fighting to get the tone control set to where I'm happy with the resulting sound. OTOH, I can dial in the OD-3's tone knob within seconds. And if I adjust the drive on the OD-3 on the fly, I can tweak the tone knob again in just a few seconds.
I think that both ODs are great when used with really clean amps, since they basically have circuits that kind of mimic the complexities of multiple gain stages in an actual amp, both WRT EQ and clipping. IMO it's also preferable to use them with a clean amp because of their bass response, which tends to sound like garbage if you try to have a dirty amp then further clip all of that low end material. I know that a popular mod for the ODR-1 is to make the bass adjustable, but IMO when that's done, you might as well just use some other type of OD that liberally removes the bass. But that's admittedly IMO/IME.