Electronic health records (EHR) often contain sensitive medical information about individual patients, posing significant limitations to sharing or releasing EHR data for downstream learning and inferential tasks. We use normalizing flows (NF), a family of deep generative models, to estimate the probability density of a dataset with differential privacy (DP) guarantees, from which privacy-preserving synthetic data are generated. We apply the technique to an EHR dataset containing patients with pulmonary hypertension. We assess the learning and inferential utility of the synthetic data by comparing the accuracy in the prediction of the hypertension status and variational posterior distribution of the parameters of a physics-based model. In addition, we use a simulated dataset from a nonlinear model to compare the results from variational inference (VI) based on privacy-preserving synthetic data, and privacy-preserving VI obtained from directly privatizing NFs for VI with DP guarantees given the original non-private dataset. The results suggest that synthetic data generated through differentially private density estimation with NF can yield good utility at a reasonable privacy cost. We also show that VI obtained from differentially private NF based on the free energy bound loss may produce variational approximations with significantly altered correlation structure, and loss formulations based on alternative dissimilarity metrics between two distributions might provide improved results.