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Weightless Neural Networks (WNNs) are a class of machine learning model which use table lookups to perform inference. This is in contrast with Deep Neural Networks (DNNs), which use multiply-accumulate operations. State-of-the-art WNN architectures have a fraction of the implementation cost of DNNs, but still lag behind them on accuracy for common image recognition tasks. Additionally, many existing WNN architectures suffer from high memory requirements. In this paper, we propose a novel WNN architecture, BTHOWeN, with key algorithmic and architectural improvements over prior work, namely counting Bloom filters, hardware-friendly hashing, and Gaussian-based nonlinear thermometer encodings to improve model accuracy and reduce area and energy consumption. BTHOWeN targets the large and growing edge computing sector by providing superior latency and energy efficiency to comparable quantized DNNs. Compared to state-of-the-art WNNs across nine classification datasets, BTHOWeN on average reduces error by more than than 40% and model size by more than 50%. We then demonstrate the viability of the BTHOWeN architecture by presenting an FPGA-based accelerator, and compare its latency and resource usage against similarly accurate quantized DNN accelerators, including Multi-Layer Perceptron (MLP) and convolutional models. The proposed BTHOWeN models consume almost 80% less energy than the MLP models, with nearly 85% reduction in latency. In our quest for efficient ML on the edge, WNNs are clearly deserving of additional attention.