What is the difference between pension investing as a self-employed person and saving?

Two differences. First the return: saved money grows with interest, invested money with market returns, which are generally higher over the long term but can also fall.

Second, and often more important for zzp'ers: with pension investing within your jaarruimte your contribution is deductible and your capital sits outside box 3. With ordinary saving it does not. In return, you can't access it before your retirement date. Read pension investing versus pension saving.