N00b question about distortion

So I apologize in advance if this is a stupid question, I'm not familiar with the specifics of bass gear.

So basically what I want to know is if distortion coming from a pedal or the amp itself will damage a cabinet? I know that if you put the gain up too much on your amp and it starts clipping, the speakers will distort and it will damage the speaker. But will that also be the case with a distortion pedal or overdrive? Or are those types of distortion completely different?

Just for reference: The reason I'm asking this is because I'm getting ready to buy an Ampeg PF-500 (with a 115HE cab). In some time I'd also be purchasing an overdrive pedal in the near future.