Embedded GLIBC (EGLIBC) is a variant of the GNU C Library (GLIBC) which is designed to work well on embedded systems. It strives to be source- and binary-compatible with GLIBC. EGLIBC's goals include a reduced footprint, configurable components, and better support for cross-compilation and cross-testing.
The Linux From Scratch project is intended for Linux users who want to build their own custom Linux system. Reasons for wanting to build such a system are diverse. Perhaps you want to get into more detail as to what happens behind the scenes. Perhaps you are fed up with the bloated standard distributions. Or perhaps you don't want to rely on pre-compiled binaries out of concerns for security.