Early identification of Cervical Cancer (CC) helps in reducing the death rate of CC patients. A Pap smear/Pap test, which is also a widely used method, is used to detect CC in its early stages. The automatic detection and categorization of Pap smear images is a difficult task. So, it is important to design and implement a high-efficiency with low-cost screening system. Therefore, this paper compares the efficiency of two deep learning-based cervical cancer screening strategies. The first technique taken for comparison is the Mutation based Atom Search Optimization (MASO) optimized DenseNet 121 architecture. The MASO technique is used to adjust hyperparameters such as learning rate value, batch sizes, and the number of neurons in the dense layer in the DenseNet 121 architecture. The second technique taken is the Sooty Tern optimized CNN-based Long Short-Term Memory (ST-CNN-LSTM) classifier to identify the different types of cervical cancer stages. The areas affected by the cancer are identified using Kernel-Weighted Fuzzy Local Information C-Means clustering (KWFLICM) model. This model removes the nucleus and cytoplasm from the background area of the cell. Normal, mild dysplastic, severe dysplastic, and carcinoma are among the CC classifications offered by these two classifiers. The efficiency of these classifiers is compared in two datasets (Hervel and SIPaKMeD) via different performance metrics such as specificity, precisions, F-score, accuracy, confusion matrix, sensitivity, and recall. The ST-CNN-LSTM offers improved performance in terms of accuracy (99.80%), sensitivity (98.83%), and specificity (99%) when compared to other classifiers.