Why are they not integral to the lower stages, e.g. like the Falcon 9? This applies not only to the ring between the first and second stage and also to the shoulder between the second and third stage.
If the angular rates on the first stage are not close to zero there is a chance of the inter-stage contacting the engine bells. (Falcon 1 flight 2?)
With a separate inter-stage the upper stage can hang onto it (protecting the engine bells) until under propulsion and rapidly accelerating away from it reducing the chance of contact.
From Stages to Saturn p. 212
The dual-plane separation was an alternative to a method called "fire in the hole," which involved ignition and separation of the S-II while still in contact with the interstage but not attached to it. Designers preferred to avoid this alternative because of possible perturbations and oscillations at the end of the first-stage boost phase. With the S-II accelerating on an even course, it was easier to drop the interstage during that phase, rather than risk hitting a wobbling interstage attached to the S-IC as the S-II pulled out.
(Picture from the Skylab Saturn V Postflight Report)
As Organic Marble's answer points out, avoiding the 'natural' simplest solution of the "fire in the hole" is a good idea. As to why this is different, its worth noting, the falcon has a "pusher" assembly to mechanically separate the stages to avoid the same problem. So its not just that one is better than the other.
What changed? Its a little hard to say exactly, in part because all such decisions are complex and interwoven with other design choices, but the most obvious is re-usability.
Pyrotechnics are light weight and pretty reliable but if you want to fly things again, cutting your rocket into ever smaller pieces is a practice best avoided.