Wow there are a lot of inaccuracies in this thread, for starters the Amiga started development in 1982 by a third party company called the Amiga Corporation and then picked up by Commodore in 1984.
Jack Tramiel the founder of commodore then resigned after a falling out with the chairman and formed his own company, he then bought a division of atari and used the name atari, he got his engineer to develop a low cost alternative to the Amiga in 1984 so the Amiga is older than the ST and superior with a much better sound chip, it had custom chips agnus with integrated blitter chip, denise and paula allowing the processor to perform other tasks.
The Amiga and Atari ST were both released in 1985, ST came out a few months earlier only because of a law suit.
Play Pacmania on the Amiga and compare it to the atari ST version, the Amiga version was full screen with great stereo sound while the ST had a small window playing area on the right of the screen with the score, lives left etc on the left with poorer sound, many games on the amiga were superior with better sound, better scrolling, more sprites, parallax scrolling, more colours etc.
The ST didn't have hardware scrolling so games developers had to employ tricks to get decent horizontal scrolling like in metrocross.
I had an ST first because in 1987 it was cheaper with a bigger games library, eventually though in 1988 the Amiga had its price cut and more games companies started to release software on it and by late 1988 i sold my ST for an Amiga 500, Amiga owners had to put up with poor ST ports not really taking advantage of the Amiga superior capabilities, read any magazine from this era and read all about this, eventually titles such as Shadows of the Beast came out to really show what the amiga was capable of, aweing people in shop windows.
Team 17 were probably the best software company in the later years of the Amiga's life really pushing the hardware to it's limits, check out Project-X, a beautiful horizontal scrolling shoot em up that the Atari ST would have really struggled with.