One of the big differentiators in the past between hard and soft bottles was that soft bottles would absorb some of the flavor from drinks.
I'm not sure how much that affects soft bottles on the market today.
In contrast to what @JustinC said, I believe hard bottles are more likely to explode. Due to their rigidity they will hold their shape until the pressure reaches a critical point then the bottle would shatter where the soft bottle would deform before reaching the critical pressure.
Also I have had hard bottles break when setting (or dropping) onto a hard surface where a soft bottle would flex.
I have heard of some hikers who will go to extreme lengths to reduce weight in their pack and they use old Gatorade/Powerade bottles since they are lighter weight than even the soft bottles. But these are much less sturdy and prone to breakage.
Most of the bottles I use are the hard variety, but I believe that is just lingering distrust of the flavor saving of the soft bottles.
I'm not sure about the prescence of BPA in soft vs hard bottles, but that should not be a problem anymore since the majority of bottles do not contain BPA any more.