arm gcc toolchain as arm-elf or arm-none-eabi, what is the difference?

Here is an excellent doc.

Tool chains have a loose name convention like arch [-vendor] [-os] – eabi

 arch -    refers to target architecture (which in our case is ARM)
 vendor -  refers to toolchain supplier
 os -      refers to the target operating system
 eabi -    refers to Embedded Application Binary Interface

some illustrations as follows :

• arm-none-eabi – This tool chain targets for ARM architecture, has no vendor, does not target an operating system and complies with the ARM EABI.
• arm-none-linux-gnueabi – This toolchain targets the ARM architecture, has no vendor, creates binaries that run on the Linux operating system, and uses the GNU EABI. It is used to target ARM-based Linux systems.