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

Question

When you build a gcc toolchain there is the possibility to build it as arm-elf or as arm-none-eabi, but what is the difference?

I use the eabi today, but that is just since everyone else seem to do that... but since that is a really bad argument, it would be really nice to understand the difference.

Note: This toolchain will crosscompile code for Cortex-M3 based mcu:s like the stm32.

Thanks

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Some links:

EABI:

• http://en.wikipedia.org/wiki/Application_binary_interface
• http://infocenter.arm.com/help/index.jsp?topic=/com.arm.doc.subset.swdev.abi/index.html

ELF:

• http://en.wikipedia.org/wiki/Executable_and_Linkable_Format