'Grilled' means cooked over a fire, by direct dry heat. It can be a wood or charcoal fire, or an electric or gas substitute. The grilled meat or fish often has black marks on the bottom from the grill rack. Grilling is a fast method; you can do it on the porch or in the back yard, on a little hibachi or a larger grill. If you have friends over for food cooked on an outdoor grill, you're cooking out. You could also say you're barbecuing, meaning cooking on a barbecue grill, but you wouldn't really call the food barbecued when cooked by this method.

Sometimes on restaurant menus, grilled fish, seafood, or chicken may just mean cooked simply and quickly, without heavy added sauces. It may not literally have been cooked on a grill, but sautéed in a pan on the stove or broiled under the broiler in the oven indoors.

'Barbecued' can mean cooked over a fire, but slowly, for hours, usually in a closed oven, not a regular kitchen oven but a big pit-style brick or stone oven, or perhaps a large, heavy, closed metal barbecue grill outdoors. A large piece of beef that has been barbecued (usually brisket) will be black all over on the outside, but soft and tender inside.

It can also mean prepared or served with barbecue sauce, which is mildly spicy. This is the sense meant by barbecued chicken, which you can make at home in the oven or on the stove.

But there are regional differences, even within the US.

A grill can also be a function in a normal household oven by which heat is proveded rapidly by elements located in the upper part of the oven, therefore above the food being cooked, whereas on a simple charcoal barbecue the heat is normally from below, unless of course it's a more sophisticated electric or gas barbecue of which there are numerous types.
To me cooking over a fire would also be a barbecue; in that context, i.e. an open fire, I'd never say grilled, and I would say "barbecued food" or "food from the barbecue".
Also, I would never describe food that has been grilled in the oven in my kitchen as barbecued. To me that is the basic difference.

To answer your other question: I would also never call something grilled if it is cooked in a frying pan - in that case I would translate gebraten with fried. If it has beengebraten in the oven, then it has been roasted.