I have not been on Pirates since they renovated it, but from what I have read, the changes are cosmetic.
There is one drop (in the dark) near the beginning of the ride. It is pretty mild. There is a pretty big splash when the boat hits the bottom, but it really doesn't feel as hard as it sounds. Disney doesn't have any warnings on it for people with back problems.
Occassinally, your boat might get hit by another if they back up. The boats don't go very fast, so they don't hit with much force, but they can bump.