I understand the formal differences between them, what I want to know is when it is more relevant to use one vs. the other. 


Do they always provide complementary insight about the performance of a given classification/detection system? 
When is it reasonable to provide them both, say, in a paper? instead of just one?
Are there any alternative (maybe more modern) descriptors that capture the relevant aspects of both ROC and precision recall for a classification system?


I am interested in arguments for both binary and multi-class (e.g. as one-vs-all) cases.

The key difference is that ROC curves will be the same no matter what the baseline probability is, but PR curves may be more useful in practice for needle-in-haystack type problems or problems where the "positive" class is more interesting than the negative class.

To show this, first let's start with a very nice way to define precision, recall and specificity.  Assume you have a "positive" class called 1 and a "negative" class called 0.  $\hat{Y}$ is your estimate of the true class label $Y$.  Then:
$$
\begin{aligned}
&\text{Precision} &= P(Y = 1 | \hat{Y} = 1)  \\
&\text{Recall} = \text{Sensitivity} &= P(\hat{Y} = 1 | Y = 1)  \\
&\text{Specificity} &= P(\hat{Y} = 0 | Y = 0)
\end{aligned}
$$
The key thing to note is that sensitivity/recall and specificity, which make up the ROC curve, are probabilities conditioned on the true class label.  Therefore, they will be the same regardless of what $P(Y = 1)$ is.  Precision is a probability conditioned on your estimate of the class label and will thus vary if you try your classifier in different populations with different baseline $P(Y = 1)$.  However, it may be more useful in practice if you only care about one population with known background probability and the "positive" class is much more interesting than the "negative" class.  (IIRC precision is popular in the document retrieval field, where this is the case.)  This is because it directly answers the question, "What is the probability that this is a real hit given my classifier says it is?". 

Interestingly, by Bayes' theorem you can work out cases where specificity can be very high and precision very low simultaneously.  All you have to do is assume $P(Y = 1)$ is very close to zero.  In practice I've developed several classifiers with this performance characteristic when searching for needles in DNA sequence haystacks.

IMHO when writing a paper you should provide whichever curve answers the question you want answered (or whichever one is more favorable to your method, if you're cynical).  If your question is: "How meaningful is a positive result from my classifier given the baseline probabilities of my problem?", use a PR curve.  If your question is, "How well can this classifier be expected to perform in general, at a variety of different baseline probabilities?", go with a ROC curve.
