Technically the difference is that a saw core charges then fully discharges the capacitor whose PD is used as output (which is the core of the oscillator and used to generate, or shaped into other waveshapes). The triangle core oscillator flip-flops the charging voltage polarity meaning the up and down ramps are both “charging” events.
Regarding a sine wave in particular, effectively all you do is filter a triangle wave. Obviously a triangle core Oscillator directly outputs a triangle wave, so let’s talk about how a saw core Oscillator is shaped into a triangle. Effectively you flip the lower half of the triangle and offset/amplify the result into the triangle shape. If the saw and triangle were “mathematically accurate” then the resulting triangles would be identical. In reality the ramp on the cores are not perfectly linear. This means that in a saw core the upper and lower half of the waves have quite different profiles. The outcome is that the triangle resulting from the saw core is less symmetric than the one produced by the triangle core. This means that the triangle or sine from a saw core are less symmetric, and further from mathematically precise than those from a triangle core.
Regarding sound, I think it can be summarised as “greater asymmetry means more harmonic content”. It’s easy to think of harmonic content as good, but the typical goal in Oscillator design is to create the mathematically prescribed waveforms. You can therefore think of the asymmetry as harmonic distortion. Harmonic distortion manifests as energy away from the fundamental. This means that for the same peak amplitude you get a “weaker fundamental”. Sonically you can think of saw as having more buzz.
As for FM, the key to understanding it is that FM creates side bands based on the harmonic content of the waves involved. The greater symmetry and lesser harmonic content of the triangle core waves reduce the harmonics in the source waves and hence the number/strength of side bands; conversely saw waves result in more harmonics in the source waves and hence side bands. This means that you get stronger side bands (and comparably weaker fundamental) in the resulting wave and less controllable harmonic content from a saw core. In other words FM from a triangle core is usually easier to tame and use musically.
As for a digital oscillator it’s trivial to make mathematically accurate waveforms. Moreover the waveforms they output are sometimes VA (meaning designed to emulate analogue oscillators): they could be “precise” if desired.
So now the question as to which is better? Well in this sense better is a matter of opinion. If the goal is mathematically precise then digital is the way. If you want lots of asymmetry and harmonics, then saw core might work; triangle sits in the middle ground. FM (linear or exponential) is another point of comparison, as is spectrum of the waveforms. In the end I think you’ll have to think about how you intend to use it (musically) and hence decide which you prefer the sound of in that context.
