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?

(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.)
