As you might already know db4o's first take on database browsing tools was called "Object Manager" or OM for short. OM was originally developed by Travis Reeder as a stand alone application and it consisted of a GUI tool and API to browse and query the contents of db4o database files (100% java code, the GUI is Swing based). The project was released under the GPL 2.0 and is currently maintained by Gisbert Avellan (a long time db4o dVP).
Object Manager Enterprise (OME), on the other hand, was conceived as a commercial tool to give premium users the ability to not only explore/query the db via a GUI but also interact with db4o's customer support portal if you have a support account (an important tool if you want to take the db4o support experience to the next level). It comes as an IDE plugin and is currently available for Eclipse and Visual Studio. (Note: OME comes with the db4o distribution starting from v 7.4 so there's no separate download package)
The good news is that, just like OM, OME has been released as open source to the community!
So, let's take a quick look at the high level features and compare OM vs OME.
OM
OME
Standalone application
IDE plugin (Eclipse, VS2005/2008)
Requires Java
Runs inside your development environment (nothing extra to install)
Should be downloaded and installed separately
Comes together with db4o installation. .NET version can be installed with the same installer
Not .NET native (runs using generic reflector)
Native both to Java and .NET
Allows to specify assemblies, which contain persistent classes definitions (no need to install in GAC to have class definitions)
SQL-like query language
Visual drag & drop query builder
Read-only indexes
Indexing within OME
Access to db4o online resources from within OME
Hope this helps you learn about the different options. Also, let's not forget forget about the great db4o browsing tools and IDE plugins that have been contributed by the community (thanks guys!)