What are the differences between the models?

First, the pickups. (American made) Jazzmasters have rather flat shallow wound pickups that will pick up a wider range of the swinging string, so the sound is unlike every other (Fender) singlecoil. It’s warmer, to some people “muddier” sounding as a normal singlecoil, but still has got the “edge” to it. Japanese Jazzmaster models have “normal” sized singlecoil pickups, thus lacking this “trademark” sound (but still sounding good). At least this is fact for the older japanese re-issue models, currently I have no info on the pickups used on newer japanese models.

Second, the scale length. All Jazzmasters have the “normal” 25.5″ (648 mm) scale length, like the Stratocaster and Telecaster models. Jaguars have the “short scale” length of 24″ (610 mm). This adds to the more “percussive” character of the Jaguar – the shorter scale length and the “normal” sized single coil pickups will result in a sharper, “tchack”-ish sound, whereas the Jazzmaster is mellower. This (the scale) is also the reason why Jaguars can be equipped with stronger string gauge than you usually would use on your “normal” scale guitar. One or even two steps up in size are ok, i.e. if you are used to .9s you can try .11s on a Jaguar – the “feel” will be nearly the same.

Third, the controls. Jazzmasters and Jaguars have the unique “rhythm circuit” design which in short enables two different sound presets which can be selected by the flip of one switch. The two independent sets of controls are on the upper horn above the strings for the “rhythm circuit” and below the strings, in the “traditional” place, for the “lead circuit” settings. Jazzmasters are equipped with a three-way toggle switch for the pickup-selection (neck, both, bridge) and two potentiometers for tone- and volume settings. Jaguars however have not the toggle switch, but a set of three slider-switches, two of which control the pickups and one adds a capacitor to the cirquit that filters low frequencies (therefore nicknamed the “strangle switch”). So from the player’s view, the switch closest to the neck controls the neck-pickup, the next switch controls the bridge pickup, and the one closest to the volume- and tone-controls is the “strangle switch”. Since the slider switches are on/off, it is possible to “mute” the Jaguar by setting both pick-up switches to “off” (usually the lower position). Players with a wide strumming area and the habit of playing close to the neck can accidentally switch the pickups “off” while playing.
The circuits on both models are designed the same, but the Jazzmaster has different component-values to compensate for the different pickups.

Fourth, the body shape is slightly different. Jazzmaster have a bit more “butt” and the upper horn is more pronounced. Both guitars have the “off set waist” design, which is really comfortable to play. The shape of the headstock differs slightly, too (see the pictures in this thread).

For more information on the details of the models, see the “history” part, an illustrated feature on the differences of the pickups can be found in the “goodies” section.