The value of adding the ROC graph if the AUC is given I always see in papers that when they want to show how they classifiers performed, they provide ROC graph and the AUC score. Now as far as I know only the AUC tells how well the classifier performed, so what is the advantage of adding the ROC graph? What can one tell from it?
 A: I usually give the ROC plot but not the AUC: For my applications it is usually clear that either a specific or a sensitive regocognition is needed. The ROC is different for classifiers that are specific but not sensitive vs. sensitive but not specific while the AUC hides this information.
Besides, one can put a whole lot of further information into the plot, e.g. color-coding the thresholds (check whether the classifier is well calibrated if the primary output is posterior probability), model stability (after resampling validation), or confidence regions for a chosen classifier (if one chooses a threshold). Finally, you can even put "extented" measures of sensitivity and specificity which do not require the thresholding @FrankHarrell fights against, e.g. it is possible to extend the concept of sensitivity and specificity and the concept behind Brier's score to yield such measures. 
A: I have not seen a single example where the graph changes our actions or the way we think.  I think the ink:information ratio in an ROC graph is enormous.  But the worst thing about it is that it tempts us to try to select a cutoff for the predicted risk, which is arbitrary, and inconsistent with optimum decision making.
Beware of the word classifier which implies discarding continuous information.
A: The ROC curve is the specificity/sensitivity plot; the AUC is the Area Under Curve. 
To be brief, the ROC curve can be interesting because it allows comparison of the sensitivity/specificity behaviour of the model. More simply:
$ROC = (x,y) \in R^2 \Rightarrow AUC = z$ but $AUC = z \nRightarrow ROC = (x,y) \in R^2$ 
A: There is great value in showing the entire ROC esp. when comparing two different classifiers, as it helps us to see whether different curves cross each other. One is not superior to the other, overall, if they cross - see the Figure 3 (screenshot shown below) from 
Seong Ho Park, J. M. G. C.-H. J. (2004). Receiver Operating Characteristic (ROC) Curve: Practical Review for Radiologists. Korean Journal of Radiology, 5(1), 11–18.

In our own study, we noticed this behaviour for biomarkers that have similar performance, and partial AUCs can be studied to demonstrate their utility for particular applications e.g. in high specificity regions for early detection of Alzheimer's disease.
Citation: Raamana, P. R., Weiner, M. W., Wang, L., & Beg, M. F. (2015). Thickness network features for prognostic applications in dementia. Neurobiology of Aging, 36, S91–S102.

Hope that helps make the case for presenting the ROC always.
