Summary (from a 1985 perspective):
Atari ST: cobbled together from mostly off-the-shelf parts, limited graphics-related capabilities, very limited inbuilt audio, nasty keyboard, nastier operating system (GUI provided by Digital Research's GEM, other OS parts via "TOS" which was a CP/M clone) with no multi-tasking.
Amiga: designed over a period of years, highly configurable video capabilities / graphics modes, highly advanced audio capabilities, fully pre-emptive multitasking operating system designed from scratch to suit the hardware (e.g. use of multiple draggable screens, etc).
The ST gained acceptance in music studios as a MIDI controller, thanks to its inbuilt MIDI interface, and also as a cheap DTP system, thanks to a dedicated monochrome 640x480 monitor (which gave a rock-solid display) and a cheap laser printer. Its 68000 was also clocked slightly faster than the Amiga's. For the early years of both machines, the ST was significantly cheaper than the Amiga.
And the Amiga story I guess we all know!
Just my 2c worth! I'm sure my description of the ST will be savaged! :-D
- Ali