While the two statistics measures are likely to be correlated, they measure different qualities of the classifier.
The area under the curve (AUC) is equal to the probability that a classifier will rank a randomly chosen positive instance higher than a randomly chosen negative example. It measures the classifiers skill in ranking a set of patterns according to the degree to which they belong to the positive class, but without actually assigning patterns to classes.
The overall accuracy also depends on the ability of the classifier to rank patterns, but also on its ability to select a threshold in the ranking used to assign patterns to the positive class if above the threshold and to the negative class if below.
Thus the classifier with the higher AUROC statistic (all things being equal) is likely to also have a higher overall accuracy as the ranking of patterns (which AUROC measures) is beneficial to both AUROC and overall accuracy. However, if one classifier ranks patterns well, but selects the threshold badly, it can have a high AUROC but a poor overall accuracy.
In practice, I like to collect the overall accuracy, the AUROC and if the classifier estimates the probability of class membership, the cross-entropy or predictive information. Then I have a metric that measures its raw ability to perform a hard classification (assuming false-positive and false-negative misclassification costs are equal and the class frequencies in the sample are the same as those in operational use - a big assumption!), a metric that measures the ability to rank patterns and a metric that measures how well the ranking is calibrated as a probability.
For many tasks, the operational misclassification costs are unknown or variable, or the operational class frequencies are different to those in the training sample or are variable. In that case, the overall accuracy is often fairly meaningless and the AUROC is a better indicator of performance and ideally we want a classifier that outputs well-calibrated probabilities, so that we can compensate for these issues in operational use. Essentially which metric is important depends on the problem we are trying to solve.