To add to what mettelus said:
There are generally 2 approaches to this in digital synth oscs:
1. The osc starts from zero (or whatever fixed point in its cycle) every time you play a note.
2. "free running" - this is done to be like analog oscs where the osc is continually running in the background and potentially every note starts at a different point in the cycle.
This also can apply to LFO's, and there may also be other random (or seemingly random) differences built into the synth, often to make it "more analog like".
I don't know about Omni.