The COVID-19 outbreak resulted in multiple waves of infections that have been associated with different SARS-CoV-2 variants. Studies have reported differential impact of the variants on respiratory health of patients. We explore whether acoustic signals, collected from COVID-19 subjects, show computationally distinguishable acoustic patterns suggesting a possibility to predict the underlying virus variant. We analyze the Coswara dataset which is collected from three subject pools, namely, i) healthy, ii) COVID-19 subjects recorded during the delta variant dominant period, and iii) data from COVID-19 subjects recorded during the omicron surge. Our findings suggest that multiple sound categories, such as cough, breathing, and speech, indicate significant acoustic feature differences when comparing COVID-19 subjects with omicron and delta variants. The classification areas-under-the-curve are significantly above chance for differentiating subjects infected by omicron from those infected by delta. Using a score fusion from multiple sound categories, we obtained an area-under-the-curve of 89% and 52.4% sensitivity at 95% specificity. Additionally, a hierarchical three class approach was used to classify the acoustic data into healthy and COVID-19 positive, and further COVID-19 subjects into delta and omicron variants providing high level of 3-class classification accuracy. These results suggest new ways for designing sound based COVID-19 diagnosis approaches.