libdrizzle is the client and protocol library for the Drizzle project. The drizzled server will use this as the protocol library, as will the client utilities and any new projects that require low-level protocol communication (like proxies). It is designed to be used for other language interfaces such as PHP extensions and with SWIG.
drizzle-php-ext is a PHP front end for the Drizzle client and protocol library (libdrizzle). It aims to provide a simple interface closely tied to the C library by minimizing interface changes, avoiding duplicate code, and not introducing any new functionality that is not PHP specific.