Soft robotic fingers provide enhanced flexibility and dexterity when interacting with the environment. The capability of soft fingers can be further improved by integrating them with tactile sensors to discriminate various textured surfaces. In this work, a flexible 3×3 fabric-based tactile sensor array was integrated with a soft, biomimetic finger for a texture discrimination task. The finger palpated seven different textured plates and the corresponding tactile response was converted into neuromorphic spiking patterns, mimicking the firing pattern of mechanoreceptors in the skin. Spike-based feature metrics were used to classify different textures using the support vector machine (SVM) classifier. The sensor was able to achieve an accuracy of 99.21% when two features, mean spike rate and average inter-spike interval, from each taxel were used as inputs into the classifier. The experiment showed that an inexpensive, soft, biomimetic finger combined with the flexible tactile sensor array can potentially help users perceive their environment better.