Both are single-reed woodwind instruments, and the reeds they use are fairly similar. The instrument shapes are similar (that is, there are saxophones shaped like most common clarinets), and although saxophones are usually made of brass, and clarinets of plastic or wood, there have been clarinets made of metal, so I’m guessing it’s not the material, at least not completely. Yet the instruments sound completely different from one another. What accounts for this?
The bore is different, for one thing. Saxophones have a conical bore, that gets progressively larger the farther down toward the bell you get. Clarinets have a cylindrical bore (except for the bell), which doesn’t taper over the length of the instrument until the bell flare.
Just looking at pictures, it looks like a clarinet is more like a tube and a soprano sax is more like cone. The neck (or whatever it is you attach the mouthpiece to) on the soprano sax looks much narrower than the clarinet. So overall the shape of a soprano sax acts more like a horn and the shape of a clarinet acts more like a flute timbre-wise.
Does that help (or make any sense whatsoever. I freely admit I’m totally guessing)?
Incidentally, this is also the difference between a trumpet and a cornet. The trumpet maintains a constant diameter for almost the entire instrument, while the cornet continually increases in diameter as you get further from the mouthpiece.
If I recall correctly, a clarinet only produces odd-numbered harmonics while a saxophone produces all harmonics. This is why saxophones has an octave key while clarinets have a register key, causing the pitch to jump by a twelfth. Whatever physical feature accounts for this probably goes a long way in explaining the differences in timbre.
The body material is a factor, too, although I’m not sure to what extent. Clarinets are made out of wood or plastic; saxes out of metal.
As was noted in the OP, this is not always true of clarinets.
Metal clarinets sound like wood if they’re any good (and some made in the 1920s and '30s were very good, ie: Selmer Paris).
The converse was true of the one plastic saxophone ever marketed, the 1950s Grafton alto. It sounded identical to brass saxes, and in any case had a brass neck.
No one yet has mentioned the straight walled chamber in clarinet mouthpieces. Saxophone mpcs today typically have similar chambers, but before the 1950s the typical sax mpc was round inside and bigger around than the neck opening, which made saxes sound a lot mellower and less edgy.
More detail than you may have been after, but a useful site - How do Woodwind Instruments Work?
That’s a great link, thanks!