This paper explores the connections between traditional Large Deformation Diffeomorphic Metric Mapping methods and unsupervised deep-learning approaches for non-rigid registration, particularly emphasizing diffeomorphic registration. The study provides useful insights and establishes connections between the methods, thereby facilitating a profound understanding of the methodological landscape. The methods considered in our study are extensively evaluated in T1w MRI images using traditional NIREP and Learn2Reg OASIS evaluation protocols with a focus on fairness, to establish equitable benchmarks and facilitate informed comparisons. Through a comprehensive analysis of the results, we address key questions, including the intricate relationship between accuracy and transformation quality in performance, the disentanglement of the influence of registration ingredients on performance, and the determination of benchmark methods and baselines. We offer valuable insights into the strengths and limitations of both traditional and deep-learning methods, shedding light on their comparative performance and guiding future advancements in the field.