Asymmetric text matching has becoming increasingly indispensable for many downstream tasks (e.g., IR and NLP). Here, asymmetry means that the documents involved for matching hold different amounts of information, e.g., a short query against a relatively longer document. The existing solutions mainly focus on modeling the feature interactions between asymmetric texts, but rarely go one step further to recognize discriminative features and perform feature denoising to enhance relevance learning. In this paper, we propose a novel adaptive feature discrimination and denoising model for asymmetric text matching, called ADDAX. For each asymmetric text pair, ADDAX is devised to explicitly distinguish discriminative features and filter out irrelevant features in a context-aware fashion. Concretely, a matching-adapted gating siamese cell (MAGS) is firstly devised to identify discriminative features and produce the corresponding hybrid representations for a text pair. Afterwards, we introduce a locality-constrained hashing denoiser to perform feature-level denoising by learning a discriminative low-dimensional binary codes for redundantly longer text. Extensive experiments on four real-world datasets from different downstream tasks demostrate that the proposed ADDAX obtains substantial performance gain over 36 up-to-date state-of-the-art alternatives.