Of all the cancers that afflict women, breast cancer (BC) has the second-highest mortality rate, and it is also believed to be the primary cause of the high death rate. Breast cancer is the most common cancer that affects women globally. There are two types of breast tumors: benign (less harmful and unlikely to become breast cancer) and malignant (which are very dangerous and might result in aberrant cells that could result in cancer). To find breast abnormalities like masses and micro-calcifications, competent and educated radiologists often examine mammographic images. This study focuses on computer-aided diagnosis to help radiologists make more precise diagnoses of breast cancer. This study aims to compare and examine the performance of the proposed shallow convolutional neural network architecture having different specifications against pre-trained deep convolutional neural network architectures trained on mammography images. Mammogram images are pre-processed in this study's initial attempt to carry out the automatic identification of BC. Thereafter, three different types of shallow convolutional neural networks with representational differences are then fed with the resulting data. In the second method, transfer learning via fine-tuning is used to feed the same collection of images into pre-trained convolutional neural networks VGG19, ResNet50, MobileNet-v2, Inception-v3, Xception, and Inception-ResNet-v2. In our experiment with two datasets, the accuracy for the CBIS-DDSM and INbreast datasets are 80.4%, 89.2%, and 87.8%, 95.1% respectively. It can be concluded from the experimental findings that the deep network-based approach with precise tuning outperforms all other state-of-the-art techniques in experiments on both datasets.