Generative speech enhancement has recently shown promising advancements in improving speech quality in noisy environments. Multiple diffusion-based frameworks exist, each employing distinct training objectives and learning techniques. This paper aims to explain the differences between these frameworks by focusing our investigation on score-based generative models and the Schrödinger bridge. We conduct a series of comprehensive experiments to compare their performance and highlight differing training behaviors. Furthermore, we propose a novel perceptual loss function tailored for the Schrödinger bridge framework, demonstrating enhanced performance and improved perceptual quality of the enhanced speech signals. All experimental code and pre-trained models are publicly available to facilitate further research and development in this domain.