When is it appropriate to deliver subtle feedback, as opposed to direct feedback to members of your team? When is a gentle nudge more effective than explicit direction?

When the feedback may be public.  Feedback is critically important for growth but the person providing the feedback needs sufficient emotional intelligence to understand when and how to provide the feedback. Even something as simple as flagging an email typo is better sent to the initial drafter instead of "replying all" and letting everyone on the chain know.   "Praise in public, criticize in private." Vince Lombardi

I think feedback is often underappreciated.  And typically the most important feedback has the quality that it will hurt the person receiving it.  Start by building a culture of feedback.  If you do not have time to do that and you value the person who will be receiving the feedback, keep it short and focus on helping them develop their positive reaction to the feedback rather than just telling them and expecting them to deal with it.  

I do not think there is a good rule of thumb to decide between which method of feedback delivery is best, unfortunately.  But if the feedback is more about the personality or the "root" of the person, subtle feedback might be better.  It is impossible to force someone to see themselves in a new light.  All we can do is suggest and give them space to change (if change is needed).