Is higher AUC always better? Let's say we measure binary classifier performance by ROC graph, and we have two separate models with distinct AUC (The Area Under the Curve) values. Is the model with the higher AUC value always better?
 A: (observed) AUC can be influenced by statistical fluctuations
The ROC Curve is usually based on a sample of real world data and taking a sample is a random process. So there is some randomness in the AUC and if you compare two ROC curves, one might be better just by chance.
A good approach is to plot the ROC together with an indication of the remaining error, for example with a 95% confidence interval. You can also compute formal  tests whether the difference between to AUCs is significant.
(Should you happen to use R, the pROC package can do both.)
A: AUC is a simplified performance measure
AUC collapses the ROC curve into a single number. Because of that a comparison of two ROC curves based on AUC might miss out on particular details that are left out in the transformation of the ROC curve into the single number.
So a higher AUC does not mean a uniform better performance.
Example of ROC curves that are better in different parts are in this image, from this question Why did meta-learning (or model stacking) underperform the individual base learners?
You can see on the right that the black curve has a larger AUC, but there is a region where it performs less good.

Related question: Determine how good an AUC is (Area under the Curve of ROC)
