Is there a proper distinction between a slow cooked meat dish labelled a casserole and one labelled stew?
And if there is a traditional distinction would it be fair to say that the distinction is no longer observed?
In today's modern culture and cooking style, the difference is likely unobserved.
I would consider a stew less constructed than a casserole, however.
While the stew would start with generally uncooked ingredients (perhaps except for browning the meat, and likely be mixed together while cooking to give a single-dish of meat, vegetables and sauce.
A casserole might include some cooked ingredients, often be more properly layered and probably not mixed while cooking.
Is there some dialectical thing going on here? I have always known stews as stovetop and casseroles as baked, just as Jared said in his comment. See for example on wikipedia: stew vs. casserole; or in Merriam-Webster: stew (click the verb form) vs. casserole. (Casserole refers to the dish as well as the food cooked in it; it's pretty definitely something you would only bake in.)
Now and then I see things get muddled when I'm reading recipes, but these are definitely the meanings I grew up with and see in the vast majority of what I read.
In addition to Joe's great answer, in my childhood (where they were usually called 'bakes' (the food), but they came out of a 'casserole' (the vessel).) I learned two other important differences:
time. A 'stew' was never done in less than 2 hours. It wasn't uncommon for it to bubble away in the crock pot or a dutch oven for 4 or 5 hours. A 'bake' by comparison was usually well under an hour in the oven and then done. If the menu called for a stew, and it wasn't already bubbling along and the kitchen cleaned up when I got home from school, I made sure I got a decent snack for myself, 'cause dinner would be a tad later than normal. :)
meal vs entree. A 'stew' was traditionally an entire meal. It might be served with some bread or roles to dredge up the last bits from your bowl, but the stew was really the entire meal. A 'bake' always had supporting dishes of some sort. Chicken noodle casserole came to the table with green beans (sometimes in a casserole of their own.) and carrots on the side, lasagna arrived with a salad and garlic bread. etc.
As to your second question, I still see the distinction in my family, my wife's family, friends and many formally trained cooks. But largely I think because I'm in the midwest, where hotdish is king, and everyone here knows what hotdish is; I see more blurring of the line between 'soup' and 'stew' than I do 'stew' and 'casserole/bake/hotdish' to be honest.
As Jared mentioned, 'stewing' always involves a fair amount of liquid -- it's a slow, moist cooking process. It's not quite a braise, as with a braise, the item is only partially submerged, whereas with a stew, you have smaller chunks of things that might float, but for the most part are submerged. You can make really thick stews (I tend to grate a potato into the stew towards the end to get it to thicken up 'til it's thick like a good gravy), but they tend to be loose, not a single congealed mass. It might be an American thing, but 'stew' in general is always assumed to have meat in it (typically a red meat, eg. beef or venison)
Casseroles (might be called a 'bake' or a 'hotdish' depending on your region) are almost always baked and untouched during the baking process. They can be layered (as with a lasagne or a shepherd's pie) but can also be just a mixture put into a baking dish and tossed in the oven (eg, my mom's tuna noodle casserole ... but she might've topped it with some extra cheese or breadcrumbs, so I guess you can consider that a layer). Casseroles tend to be less 'soupy' than a stew -- some will hold their own shape when they're scooped, if let to cool down for a few minutes after they come out of the oven. (eg, lasagne, most macaroni & cheese casseroles)
... as it looks like you're from the UK, it's possible that these distinctions aren't made over there. If the terms are used differently outside the US, please add to the Translating Cooking Terms post.
One factor not mentioned in the other answers is that there is a class of cookware called a "casserole dish" -- ceramic or pyrex, somewhat shallow, often with a lid. I suppose it follows that a casserole is the kind of dish you prepare in such a vessel.
I went to the dictionary for definitons of casserole and stew. As pointed out above, the main difference and the description, lies in the cooking vessel used. A casserole is a covered oven based dish whilst a stew is commonly stove-top, covered or not, using more liquid to cook the ingredients.
Slow cookers arrived on the market. I supposed these would be closer to the casserole definition?