Internet of Underwater Things (IoUTs) deals in a resource-constrained environment and has several open issues, challenges, and potential applications in both onshore and offshore fields. The distinctive features of acoustic medium and persistent node mobility have spurred the development of a routing protocol for IoUT that ensures the efficient data transfer at the surface station. The efficient data transfer metrics can be achieved either by the emergence of cluster-based or chain-based routing protocols. This paper provides a simulation-based quantitative analysis of the most recent and prominent cluster-based and chain-based routing protocols for IoUT that are intended to enable the efficient data transfer from source to destination. In this context, we use NS3 to conduct extensive simulations for the quantitative analysis of different routing protocols relating to packet delivery rate, packet drop rate, normalized energy consumption, normalized network lifetime, density of alive nodes, and density of dead nodes. In addition to quantitative analysis, we also provide the performance trade-offs, limitations, and applications for IoUT routing protocols for these two classifications. In last, we also perform quantitative analysis for these two classifications head-to-head. This work aims to provide helpful insights into selecting a suitable protocol for routing applications to meet the various specifications and requirements of IoUT for efficient data dissemination.