Yes, pots are the difference between the two circuits.
The treble side of the Jaguar & Jazzmaster uses 2 1MEG Ohm Audio Pots, whereas the bass side of the JAguar and Jazzmaster have a 50K Linear Pot and a 1MEG ohm Linear pot.
A potentiometer is a variable resistor, what that means is that the resistant value on a control pot is varied by resistive material inside it. There is a "wiper" contact n the center lug of a potentiometer that sweeps around on that "resistive track". The ends of the resistive coating are connected to two other lugs that are the outer two. Anyway, if you turn the pot one way, it gives the one side of the resistive track increasingly more resistance, and the other side less. However, the type of material and the way it's coated on a poteniometer makes a difference on weather the change is considered "Linear" or "Analog/Audio". Linear pots seem to sound more like a gradual change than an audio pot, and also, from my experience, sound warmer and block out treble frequencies more.
The rhythm circuit sounds darker and warmer for the fact that that was what it was designed for, however, I break the rules and use it for "Woman" tone.
As for pots. The lead circuit uses 1MEG Audio pots, which were designed for audio applications, have a more extreme change in the in-between settings, and tend to become more bright and clear as the volume is turned up. 1MEG pots seem to be unique to the Jaguar and Jazzmaster from my experience, most Fender style guitars use 250K Audio pots, and most Gibson style guitars use 500K. The reason for that is that 250K bleeds off less treble than a 500K does, and Fenders typically use smaller single coils which are thinner sounding pickups, so the 250K is to compensate for the lack of "beef" to the bottom end. They switched to 1MEG on the Jazzmaster to brighten up the big fat coils, and kept it on the Jaguar because a thinner twang was a more popular sound in those days.
Basically, the whole idea on the Rhythm/Lead circuit design was to have one circuit sound dark and warm for rhythm work, as most guitar playing in those days was done with a dark/warm setting for rhythm, for instance, that's why most Gibsons had a separate set of volume and tone controls for the neck pickup and bridge pickup, because Jazz Guitarists would turn the tone down and the volume on the neck a little, and use that for rhythm, then throw the guitar into the bridge pickup for leads, usually just a little short of flat out. It was the "fender" way of making a Jazz guitar initially, and the surf guys liked it better but wanted some improvements...boom, enter the Jaguar.
Oddly enough, I've never used my Rhythm circuit for rhythm, at least, not that much, I end up using it more for that Eric Claptop type "Woman Tone", lol.