Online tuning of particle accelerators is a complex optimisation problem that continues to require manual intervention by experienced human operators. Autonomous tuning is a rapidly expanding field of research, where learning-based methods like Bayesian optimisation (BO) hold great promise in improving plant performance and reducing tuning times. At the same time, reinforcement learning (RL) is a capable method of learning intelligent controllers, and recent work shows that RL can also be used to train domain-specialised optimisers in so-called reinforcement learning-trained optimisation (RLO). In parallel efforts, both algorithms have found successful adoption in particle accelerator tuning. Here we present a comparative case study, assessing the performance of both algorithms while providing a nuanced analysis of the merits and the practical challenges involved in deploying them to real-world facilities. Our results will help practitioners choose a suitable learning-based tuning algorithm for their tuning tasks, accelerating the adoption of autonomous tuning algorithms, ultimately improving the availability of particle accelerators and pushing their operational limits.