Tes4Mod:Bash vs. OBMM

Previously, Wrye Bash and OBMM (Oblivion Mod Manager) overlapped primarily in mod ordering functions, hence OBMM was recommended for mod installation, and Bash for the other eighty or so functions that it provides. However, Wrye recently (January 2009) added a mod installer component to Bash that competes directly with OBMM's mod installation feature. 

Bain Overview
Bain (Bash Installers) is a mod package installer built into Wrye Bash. Bain is specifically designed to avoid some of the difficulties associated with using OBMM, however it has also left out some OBMM features (notably BSA editing/viewing). 
For general documentation of Bain, see Wrye Bash: Installers Tab. This documentation is very thorough with a "Getting Started" guide, description of various features, etc.  
Bain is optional for Wrye Bash users. Those who prefer to install mods manually or use OBMM and still use the other features of Wrye Bash can do so with no problems.

Bain Advantages
No OMODS required. 
Just drop existing zip, 7z and rar archives into the Oblivion Mods\Bash Installers directory, and Bain will immediately display them as an available package. Bain will also recognize directories of files (i.e. uncompressed archives) as valid packages.
Bain packages can readily be opened in regular packages viewers. Just choose "Open" from the package's context menu.
Ordered Install/Uninstall: 
Bain assigns an install order to all packages (which players can easily change as desired). As with mods, later (higher order) packages trump earlier (lower order) packages.
When a package is installed, it will override files installed by lower order packages, but not files already installed by higher order packages.
Similarly when a package is uninstalled, an "annealing" process will automatically reinstall files from lower level packages that were previously obscured.
Useful Status Info:
Bain provides a lot of readily available status information. The degree of a package's synchronization with the oblivion\data directory is shown through the icon color (independent of whether the package is installed or not), any information regarding missing files, mismatched files, etc. is readily available through detail tabs. In particular the override tab will tell you which other packages are overriding the files from the current package.
OBMM also offers status info, but the info that it offers is not as thorough and readily accessible as with Bain. (E.g. the icon color for an active package in OBMM is simply blue -- even if files from that package have been subsequently overridden or removed; while in Bain that synchronization info is always shown correctly by icon color.)
Document Sweeping
Bain sweeps document type files (.txt, .rtf, etc.) into the Oblivion\Data\Docs directory. It will also rename "readme.txt" doc files during the sweep process to ensure that they have unique names and won't be overwritten by other mods.
Project Building
Bain has several features that are designed to make it very easy for modders to create mod archives. Archives can be decompressed readily into directories, which are then packages in their own right. Directory (aka "Project") packages can then be synchronized to changes in the data directory (i.e., edit your mod in the data directory, then sync to the project directory and then archive your project directory, and you're done).

Bain Disadvantages
Bain does not support arbitrary installation scripts.  This could be seen as an advantage, though, since there is 0 chance that a BAIN installation wizard will hose your system.
No shader merging (yet)
Bain does not provide viewing, packing or unpacking of BSA packages (yet)
Note: Bain does provide a BSA-Redirection mechanism.
Bain cannot scan packages to determine if meshes and/or textures are missing. (yet)