Artificial intelligence (AI) can extract visual information from histopathological slides and yield biological insight and clinical biomarkers. Whole slide images are cut into thousands of tiles and classification problems are often weakly-supervised: the ground truth is only known for the slide, not for every single tile. In classical weakly-supervised analysis pipelines, all tiles inherit the slide label while in multiple-instance learning (MIL), only bags of tiles inherit the label. However, it is still unclear how these widely used but markedly different approaches perform relative to each other. We implemented and systematically compared six methods in six clinically relevant end-to-end prediction tasks using data from N=2980 patients for training with rigorous external validation. We tested three classical weakly-supervised approaches with convolutional neural networks and vision transformers (ViT) and three MIL-based approaches with and without an additional attention module. Our results empirically demonstrate that histological tumor subtyping of renal cell carcinoma is an easy task in which all approaches achieve an area under the receiver operating curve (AUROC) of above 0.9. In contrast, we report significant performance differences for clinically relevant tasks of mutation prediction in colorectal, gastric, and bladder cancer. In these mutation prediction tasks, classical weakly-supervised workflows outperformed MIL-based weakly-supervised methods for mutation prediction, which is surprising given their simplicity. This shows that new end-to-end image analysis pipelines in computational pathology should be compared to classical weakly-supervised methods. Also, these findings motivate the development of new methods which combine the elegant assumptions of MIL with the empirically observed higher performance of classical weakly-supervised approaches. We make all source codes publicly available at https://github.com/KatherLab/HIA, allowing easy application of all methods to any similar task.