A fundamental feature of learning in animals is the "ability to forget" that allows an organism to perceive, model and make decisions from disparate streams of information and adapt to changing environments. Against this backdrop, we present a novel unsupervised learning mechanism ASP (Adaptive Synaptic Plasticity) for improved recognition with Spiking Neural Networks (SNNs) for real time on-line learning in a dynamic environment. We incorporate an adaptive weight decay mechanism with the traditional Spike Timing Dependent Plasticity (STDP) learning to model adaptivity in SNNs. The leak rate of the synaptic weights is modulated based on the temporal correlation between the spiking patterns of the pre- and post-synaptic neurons. This mechanism helps in gradual forgetting of insignificant data while retaining significant, yet old, information. ASP, thus, maintains a balance between forgetting and immediate learning to construct a stable-plastic self-adaptive SNN for continuously changing inputs. We demonstrate that the proposed learning methodology addresses catastrophic forgetting while yielding significantly improved accuracy over the conventional STDP learning method for digit recognition applications. Additionally, we observe that the proposed learning model automatically encodes selective attention towards relevant features in the input data while eliminating the influence of background noise (or denoising) further improving the robustness of the ASP learning.