Just spotted this while perusing the forum, and I have to disagree.
This won't be absolutely authoritative by any means, as my interpretation of these throws comes from my style of jujutsu as well as my research into translations and interpretations of the original Kodokan techniques. Importantly, this comes flavoured with the view from my style that a throw is ultimately defined by the effect that it has on uke far more than on specific positioning of tori.
As a result of this, I think the comments about hand positions are misleading. Someone shorter than their opponent can do koshi-guruma or o-goshi with arms around the waist, someone taller can do both throws with arms around the neck or down the back.
Where a difference for tori really comes in is the foot positioning and the degree of hip insertion. Koshi guruma is, for me, the deepest turn and therefore the deepest hip insertion of all of the hip throws, with my foot position getting close to perpendicular with uke's, and uke almost folds directly over my hips when I do this as a result. This very deep turn has an impact on the balance take at the start, which needs a degree of sideways movement that I don't normally expect to need for o-goshi.
Going back to o-goshi, the difference starts at the balance take - less (no) sideways, more forwards. The entry then leads to an almost parallel stance to uke's, with hips extended enough to block uke's natural step. Rather than simply rolling over tori's hips, uke should feel like they are travelling a path from tori's extended hip upwards to tori's rear shoulder, then over into a throw.
Once you start identifying differences between these throws you can then start picking up on other similarities. For example, there are definite overlaps between o-guruma and koshi-guruma that do not exist when considering o-goshi instead, and understanding these commonalities more is really helping me deepen my understanding of the core principles of the throws that make up my art.