Nonlinear acoustic testing techniques have shown great potential for identification of volumetric microcracking and early damage in diverse materials. In this paper, we compare the results of two impact-based techniques: Impact-based Nonlinear Resonant Acoustic Spectroscopy (INRAS) and Dynamic Acousto-Elastic Testing (IDAET) in monitoring damage in concrete due to Alkali-Silica Reaction (ASR) and freeze-thaw (FT) cycles. Using an impact hammer as the large-strain (strain ~ 10-6 -10-5 ) source (instead of a piezoelectric ceramic or a shaker used in conventional testing) allows testing large samples and enables field transportability. INRAS gives a global measure of sample nonlinearity while IDAET provides a local but comprehensive picture of material nonlinear properties. We propose two new data processing approaches for a single-impact INRAS that while being simpler, yields similar results to those from other analyses. We then introduce IDAET and show how to extract both classical and non-classical nonlinear parameters from the test results. INRAS (various analysis approaches) and IDAET are used to monitor a set of concrete samples undergoing accelerated ASR and FT cycles. Nonlinear parameters extracted from the two tests show good agreement; all exhibiting far more sensitivity to distributed FT damage than standard resonance frequency measurements.