Medical imaging techniques, especially Magnetic Resonance Imaging (MRI), are accepted as the gold standard in the diagnosis and treatment planning of neurological diseases. However, the manual analysis of MRI images is a time-consuming process for radiologists and is prone to human error due to fatigue. In this study, two different Deep Learning approaches were developed and analyzed comparatively for the automatic detection and classification of brain tumors (Glioma, Meningioma, Pituitary, and No Tumor). In the first approach, a custom Convolutional Neural Network (CNN) architecture named "OkanNet", which has a low computational cost and fast training time, was designed from scratch. In the second approach, the Transfer Learning method was applied using the 50-layer ResNet-50 [1] architecture, pre-trained on the ImageNet dataset. In experiments conducted on an extended dataset compiled by Masoud Nickparvar containing a total of $7,023$ MRI images, the Transfer Learning-based ResNet-50 model exhibited superior classification performance, achieving $96.49\%$ Accuracy and $0.963$ Precision. In contrast, the custom OkanNet architecture reached an accuracy rate of $88.10\%$; however, it proved to be a strong alternative for mobile and embedded systems with limited computational power by yielding results approximately $3.2$ times faster ($311$ seconds) than ResNet-50 in terms of training time. This study demonstrates the trade-off between model depth and computational efficiency in medical image analysis through experimental data.