The interaction of cold atmospheric plasma (CAP) with biotargets is accompanied by chemical reactions on their surfaces and insides, and it has great potential as an anticancer approach. This study discovers the molecular mechanisms that may explain the selective death of tumor cells under CAP exposure. To reach this goal, the transcriptional response to CAP treatment was analyzed in A549 lung adenocarcinoma cells and in lung-fibroblast Wi-38 cells. We found that the CAP treatment induced the common trend of response from A549 and Wi-38 cells—the p53 pathway, KRAS signaling, UV response, TNF-alpha signaling, and apoptosis-related processes were up-regulated in both cell lines. However, the amplitude of the response to CAP was more variable in the A549 cells. The CAP-dependent death of A549 cells was accompanied by DNA damage, cell-cycle arrest in G2/M, and the dysfunctional response of glutathione peroxidase 4 (GPx4). The activation of the genes of endoplasmic reticulum stress and ER lumens was detected only in the A549 cells. Transmission-electron microscopy confirmed the alteration of the morphology of the ER lumens in the A549 cells after the CAP exposure. It can be concluded that the responses to nuclear stress and ER stress constitute the main differences in the sensitivity of tumor and healthy cells to CAP exposure.