The uses will present themselves, or, at least, that's what I've found with my chinois, which serves a similar purpose.
Whenever there are fibrous bits that you don't want to blender-ize or process, or something that is supposed to be smooth, turns out lumpy..you can reach for it.
And, of course, when you have the energy, all the creamy soups are lovely sieved. In a hurry, I still use the immersion blender, but now you have the option to refine.And you don't have to peel the apples or pears before you sauce them...which gives them richer color and taste.
You'll be glad you got it.

I made another batch of gazpacho today and WOW was it wonderful to be able to use the food mill instead of a spatula and fine-mesh strainer, I tell you what.

I use my food mill for a lot of things, but especially roasted vegetable soups (butternut squash with potato and apple is perfect for fall) and vegetable purees (I like 2/3 celeriac with 1/3 potato, and add milk and butter). I really like the texture that comes from it, as opposed to the blender/processor.

I agree with AWB, stuff that's ground in a food mill (or chinois or mortar) comes out with a more pleasing texture (and, I think, better flavor) than stuff that's sliced up by a blender or food processor. Plus, not so damn noisy!

Mashed potatoes of ANY variety are simply incredible run through the food mill. I have my mum's old one, which only has one size of hole to push stuff through, maybe 2/32th's of an inch or so. Anyhoo, the resultant mashies are fluffy without having been whipped, & still have a density that manages to be creamy without having first been reduced to a state of glue. Very delicious with cottage cheese mixed in & white truffle oil.