We focus on the problem of estimating the change in the dependency structures of two p-dimensional Gaussian Graphical models (GGMs). Previous studies for sparse change estimation in GGMs involve expensive and difficult non-smooth optimization. We propose a novel method, DIFFEE for estimating DIFFerential networks via an Elementary Estimator under a high-dimensional situation. DIFFEE is solved through a faster and closed-form solution that enables it to work in large-scale settings. Notice that GGM assumes data are generated from a Gaussian distribution. However, the Gaussian assumption is too strict and can not be satisfied with all the real-world data generated from a complex process. Therefore, we further extend DIFFEE to NPN-DIFFEE by assuming that data are drawn from the nonparanormal distribution (a large family of distributions) instead of a multivariate Gaussian distribution. Thus, NPN-DIFFEE is applicable to more general conditions. We conduct a rigorous statistical analysis showing that surprisingly DIFFEE achieves the same asymptotic convergence rates as the state-of-the-art estimators that are much more difficult to compute. Our experimental results on multiple synthetic datasets and one real-world data about brain connectivity show strong performance improvements over baselines, as well as significant computational benefits.