Both seem to try making it simpler using a database in PHP. Both seem to provide an abstraction over different database types like MySQL, SQLite, etc.
What are the differences between both ADOdb and PDO?
PDO is standard in PHP as of version 5.1. (It is also available with a PECL extension in PHP 5.0) Most hosting provides will have it enabled. AdoDB is not a standard extension.
Also, I believe the PDO drivers are "PHP-native": they are built on top of the same libraries that PHP itself was built on, and use the same underlying routines for things like memory management. So potentially, PDO is more lightweight than AdoDB.
According to this benchmark, AdoDB is considerably slower than PDO: (fixed link) https://gist.github.com/tony-landis/31483
Of course, you should consider whether this is important enough for your use case to prefer PDO or not.