According to a 2020 WHO report, cancer is one of the main causes of deaths worldwide. Among these deaths, lung and colon cancer collectively responsible for nearly 2.735 million deaths.So, detection and classification of lung and colon cancer is one of the utmost priority research areas in the field of biomedical health informatics. In this article, comparative analysis of two feature extraction methodologies has been presented for lung and colon cancer classification. In one approach, six handcrafted features extraction techniques based on colour, texture, shape and structure are presented. Gradient Boosting (GB), SVM-RBF, Multilayer Perceptron (MLP) and Random Forest (RF) classifiers with handcrafted features are trained and tested for lung and colon cancer classification. In another approach, using the notion of transfer learning, seven deep learning frameworks for deep feature extraction from lung and colon cancer histopathological images are presented. The extracted deep features (as input attributes) are applied into conventional GB, SVM-RBF, MLP and RF classifiers for lung and colon cancer classification. However, in contrast to handcrafted features a significant improvement in classifiers performance is observed with features extracted by deep CNN networks. It has been found that the proposed technique obtained excellent results in terms of accuracy, precision, recall, F1 score and ROC-AUC. The RF classifier with DenseNet-121 extracted deep features can identify the lung and colon cancer tissue with an accuracy and recall of 98.60%, precision of 98.63%, F1 score of 0.985 and ROC-AUC of 01.