In this paper, we propose an application-aware intelligent load balancing system for high-throughput, distributed computing, and data-intensive science workflows. We leverage emerging deep learning techniques for time-series modeling to develop an application-aware predictive analytics system for accurately forecasting GridFTP connection loads. Our solution integrates with a major U.S. CMS Tier-2 site; we use a real dataset representing 670 million GridFTP transfer connections measured over 18 months to drive our predictive analytics solution. First, we perform extensive analysis on this dataset and use the connection loads as an example to study the temporal dependencies between various user-roles and workflow memberships. We use the analysis to motivate the design of a gated recurrent unit (GRU) based deep recurrent neural network (RNN) for modeling long-term temporal dependencies and predicting connection loads. We develop a novel application-aware, predictive and intelligent load balancer, APRIL, that effectively integrates application metadata and load forecast information to maximize server utilization. We conduct extensive experiments to evaluate the performance of our deep RNN predictive analytics system and compare it with other approaches such as ARIMA and multi-layer perceptron (MLP) predictors. The results show that our forecasting model, depending on the user-role, performs between 5.88%-92.6% better than the alternatives. We also demonstrate the effectiveness of APRIL by comparing it with the load balancing capabilities of an existing production Linux Virtual Server (LVS) cluster. Our approach improves server utilization, on an average, between 0.5 to 11 times, when compared with its LVS counterpart.