TT rumble versus "oscillation": what's the difference?

Playing some LP's I've described forward and back movement of my mid/woofer drivers.  No obvious noise emanates from the movement that can be visualized.  I assumed that this was evidence of "rumble" but have been informed that it might instead be "oscillations".

So my question to you vinyl gurus: what is the difference between those two conditions?  Is one worse than the other?  Do both compromise performance of the speaker and the driving amplifier?

Thanks!