Traditional deepfake detectors have dealt with the detection problem as a binary classification task. This approach can achieve satisfactory results in cases where samples of a given deepfake generation technique have been seen during training, but can easily fail with deepfakes generated by other techniques. In this paper, we propose DiffFake, a novel deepfake detector that approaches the detection problem as an anomaly detection task. Specifically, DiffFake learns natural changes that occur between two facial images of the same person by leveraging a differential anomaly detection framework. This is done by combining pairs of deep face embeddings and using them to train an anomaly detection model. We further propose to train a feature extractor on pseudo-deepfakes with global and local artifacts, to extract meaningful and generalizable features that can then be used to train the anomaly detection model. We perform extensive experiments on five different deepfake datasets and show that our method can match and sometimes even exceed the performance of state-of-the-art competitors.