Model observers are computational tools to evaluate and optimize task-based medical image quality. Linear model observers, such as the Channelized Hotelling Observer (CHO), predict human accuracy in detection tasks with a few possible signal locations in clinical phantoms or real anatomic backgrounds. In recent years, Convolutional Neural Networks (CNNs) have been proposed as a new type of model observer. What is not well understood is what CNNs add over the more common linear model observer approaches. We compare the CHO and CNN detection accuracy to the radiologist's accuracy in searching for two types of signals (mass and microcalcification) embedded in 2D/3D breast tomosynthesis phantoms (DBT). We show that the CHO model's accuracy is comparable to the CNN's performance for a location-known-exactly detection task. However, for the search task with 2D/3D DBT phantoms, the CHO's detection accuracy was significantly lower than the CNN accuracy. A comparison to the radiologist's accuracy showed that the CNN but not the CHO could match or exceed the radiologist's accuracy in the 2D microcalcification and 3D mass search conditions. An analysis of the eye position showed that radiologists fixated more often and longer at the locations corresponding to CNN false positives. Most CHO false positives were the phantom's normal anatomy and were not fixated by radiologists. In conclusion, we show that CNNs can be used as an anthropomorphic model observer for the search task for which traditional linear model observers fail due to their inability to discount false positives arising from the anatomical backgrounds.