Federated Distillation (FD) is a popular novel algorithmic paradigm for Federated Learning, which achieves training performance competitive to prior parameter averaging based methods, while additionally allowing the clients to train different model architectures, by distilling the client predictions on an unlabeled auxiliary set of data into a student model. In this work we propose FedAUX, an extension to FD, which, under the same set of assumptions, drastically improves performance by deriving maximum utility from the unlabeled auxiliary data. FedAUX modifies the FD training procedure in two ways: First, unsupervised pre-training on the auxiliary data is performed to find a model initialization for the distributed training. Second, $(\varepsilon, δ)$-differentially private certainty scoring is used to weight the ensemble predictions on the auxiliary data according to the certainty of each client model. Experiments on large-scale convolutional neural networks and transformer models demonstrate, that the training performance of FedAUX exceeds SOTA FL baseline methods by a substantial margin in both the iid and non-iid regime, further closing the gap to centralized training performance. Code is available at github.com/fedl-repo/fedaux.