This is where Mod Organizer comes into its own. Many of the tools that we need were designed to work with the flat directory structure of vanilla Morrowind, not with each mod in its own folder.
Surprisingly, the most important feature is actually this ability to launch external tools, for reasons that will be explained below. This provides a helpful GUI to install, view and organise mods, without needing to manually edit OpenMW's configuration files. Mod Organiser is a mod management tool, much like Wrye Mash. However, it is still possible for conflicts to arise between mods, and this is where a few tools come in handy. OpenMW simply needs to be configured to point to the appropriate directories, as documented here. In OpenMW, each mod resides in its own directory, so no files are ever overwritten. This results in the game directory becoming polluted and mods frequently overwriting each other's files, thus necessitating the use of external tools to manage mods. In the original game, mods are all installed to the same directory.
OpenMW makes mod management somewhat easier than the vanilla engine, but it is still no easy feat. My guide for the vanilla engine - which goes into much more detail - can be found here. This guide is for OpenMW and assumes a basic knowledge of mod management.