musl is an implementation of the C standard library built on top of the Linux system call API, including interfaces defined in the base language standard, POSIX, and widely agreed-upon extensions. musl is lightweight, fast, simple, free, and strives to be correct in the sense of standards-conformance and safety.
BusyBox combines tiny versions of many common UNIX utilities into a single small executable. It provides replacements for most of the utilities you usually find in GNU fileutils, shellutils, etc. The utilities in BusyBox generally have fewer options than their full-featured GNU cousins; however, the options that are included provide the expected functionality and behave very much like their GNU counterparts. BusyBox provides a fairly complete environment for any small or embedded system.
BusyBox has been written with size-optimization and limited resources in mind. It is also extremely modular so you can easily include or exclude commands (or features) at compile time. This makes it easy to customize your embedded systems. To create a working system, just add some device nodes in /dev, a few configuration files in /etc, and a Linux kernel.