You can understand the relationship between REX and BREW / BrewMP by way of an (admittedly not perfect) analogy with the most popular desktop environment: the Microsoft Windows XP/Vista/7 operating system and the Microsoft .NET application framework.

.NET framework is a modern set of libraries, user interface controls, and data management APIs. .NET is platform-and language-independent (it can run on Windows and Linux, and supports multiple languages - C#, C++, Visual Basic, Ruby etc.). In this environment, .NET is not your OS; Windows is the operating system on the computer, and the .NET framework runs on top of the Windows OS, using Win32 APIs internally in its implementation. .NET is higher-level and easier to program for in comparison with mostly C-based, unmanaged Win32 API.

In a similar fashion, REX OS is a real-time operating system used by Qualcomm, with BREW / BrewMP available as an application framework. Just like in Windows, you can write applications against the native REX APIs, and you can use the BREW APIs. The BREW framework is higher-level, supports multiple languages, is binary backward-compatible, has a decent simulator, and offers a UI controls package. So on a BREW phone, REX is the "real" operating system, but most if not all apps are using BREW APIs and run within the BREW framework.

This chart represents the relationship well - you can think of REX as being below the "Handset APIs".

BREW apps also have code signing, distribution, and certification processes that make it easier for application writers to support multiple devices and operators. If you would like to get your REX program on a phone, you will have to talk with the OEM directly, as it would become a part of, and would have to be installed with, the operating system.