Here are some links to compare the Kitchenaid Platinum Collection mixer, Kitchenaid Artisan mixer and the Breville BEM800.
Kitchenaid Platinum Collection KSM156 - Candy Apple Red + Information
Kitchenaid Platinum Collection KSM156 - Other colours (Metallic Colours in this collection)
Kitchenaid Artisan Range KSM150 - Empire Red + Information
Kitchenaid Artisan Range KSM150 - Other colours
Breville Scraper Pro BEM800 - Silver
Breville Scraper Pro BEM800CB - Cranberry Red
Basically, the Breville is more technologically advanced with the LED light to indicate the power range, timer, load sensing technology, lift assist handles and includes a spatula.
It all comes down to personal preference with me as I'd prefer the retro, metallic red colour and manual-manner of the Kitchenaid mixer.