Searching is fundamentally different than with the eDonkey2000; Overnet uses a dedicated thread to publish a user's shared files index across the network. The published files are then propagated to the rest of the clients (similiar to other P2P systems).
The file transfer system is the same as eDonkey2000's and therefore also has all it's features, most notably the ability to share parts of files that are still being downloaded, which greatly improves the spreading and the availability of large files.
On the Overnet homepage you will find the official Windows version with a GUI, and command line interface-versions for Linux and MacOSX. On both Linux and MacOSX, you can use ed2k-gtk-gui, a third party open source gui controller wich is able to control the Overnet and eDonkey2000 command line clients). For Linux you may also try Mldonkey.
Probably the most controversial aspect of Overnet was that it was written using the .NET framework from Microsoft. This framework allows applications to be prototyped and built much more quickly than using the classic MFC platform. Unfortunately many people associate the .NET programming APIs with Microsoft's Hailstorm and Passport initiatives. However, the Overnet core has been written in C++, and the developers of Overnet have made efforts to remove the dependency of the Windows client on the .NET framework.
Since version 0.41 (December 2002) Overnet is .NET-free, so it should not be an issue anymore.