@margo,
rosti/roesti are more of a shredded potato dish, while home fries (as I know them) are more like the bratkartoffeln that Thomas knows. There are some weirdnesses with that definition as well, as some people think that home fries should be wedges or chunks of potato, while bratkartoffeln (in my world) are sliced potatoes.
nowwwwwwwwwwwww, the next question - should the potatoes be raw or cooked when they're made into home fries/rosti/bratkartoffeln?
rosti - raw fer sure
bratkartoffeln - your choice
home fries - I'd start with parboiled potatoes
(and then we get to the topic one of my favourite ever Abuzz threads - smooth or lumpy mashed potatoes?)
and those rosti - shredded isn't grated, because then you'd be making latkes/potato pancakes