This paper addresses acoustic vehicle counting using one-channel audio. We predict the pass-by instants of vehicles from local minima of clipped vehicle-to-microphone distance. This distance is predicted from audio using a two-stage (coarse-fine) regression, with both stages realised via neural networks (NNs). Experiments show that the NN-based distance regression outperforms by far the previously proposed support vector regression. The $ 95\% $ confidence interval for the mean of vehicle counting error is within $[0.28\%, -0.55\%]$. Besides the minima-based counting, we propose a deep learning counting that operates on the predicted distance without detecting local minima. Although outperformed in accuracy by the former approach, deep counting has a significant advantage in that it does not depend on minima detection parameters. Results also show that removing low frequencies in features improves the counting performance.