The next generation of mobile networks (NextG) will require careful resource management to support edge offloading of resource-intensive deep learning (DL) tasks. Current slicing frameworks treat all DL tasks equally without adjusting to their high-level objectives, resulting in sub-optimal performance. To overcome this, we propose SEM-O-RAN, a semantic and flexible slicing framework for computer vision task offloading in NextG Open RANs. Our framework accounts for the semantic nature of object classes as well as the level of data quality to optimally tailor data compression and minimize the usage of networking and computing resources. In fact, we show that different object classes tolerate different levels of image compression while preserving detection accuracy. To address the above issues, we first present the mathematical formulation of the Semantic Flexible Edge Slicing Problem (SF-ESP), which turns out to be NP-hard. We thus define a greedy algorithm to solve it efficiently, which is also able to always select the resource allocation that yields the best resource utilization, whenever multiple allocations satisfy the DL task requirements. We evaluate SEM-O-RAN's performance through extensive numerical analysis and real-world experiments on the Colosseum testbed, considering state-of-the-art computer-vision tasks and DL models. The obtained results demonstrate that SEM-O-RAN allocates up to 169% more tasks and obtains 52% higher revenues than the state of the art.