A bin file is just the bits and bytes that go into the rom or a particular address from which you will run the program. You can take this data and load it directly as is, you need to know what the base address is though as that is normally not in there.
An elf file contains the bin information but it is surrounded by lots of other information, possible debug info, symbols, can distinguish code from data within the binary. Allows for more than one chunk of binary data (when you dump one of these to a bin you get one big bin file with fill data to pad it to the next block). Tells you how much binary you have and how much bss data is there that wants to be initialised to zeros (gnu tools have problems creating bin files correctly).
The elf file format is a standard, arm publishes its enhancements/variations on the standard. I recommend everyone writes an elf parsing program to understand what is in there, dont bother with a library, it is quite simple to just use the information and structures in the spec. Helps to overcome gnu problems in general creating .bin files as well as debugging linker scripts and other things that can help to mess up your bin or elf output.
I just want to correct a point here. ELF file is produced by the Linker, not the compiler.
The Compiler mission ends after producing the object files (*.o) out of the source code files. Linker links all .o files together and produces the ELF.
some resources:
ELF format is generally the default output of compiling. if you use GNU tool chains, you can translate it to binary format by using objcopy, such as:
arm-elf-objcopy -O binary [elf-input-file] [binary-output-file]
or using fromELF utility(built in most IDEs such as ADS though):
fromelf -bin -o [binary-output-file] [elf-input-file]
Source: Stackoverflow.com