Can anyone shed some light as to why YouTube videos are being cropped
I think you are confused about few things at once. I have seen a lot of people asking the same youtube question in here and almost every time it's because their HDTVs are set on 16x9 rather than "Screen Fit" (Samsung) or "Just Scan" (LG), etc. It is always the first instinct to think Apple TV is cropping but 99% of the time it has nothing to do with Apple TV.
Let's get few things clear though because some things you said doesn't add up.
If you have a widescreen HDTV (16x9 aspect ratio):
1) The reason most blu-ray movies show black bars on top and bottom is because the movies are shot with a wider aspect ratio than 16x9. It's called Anamorphic or scope and it can range anywhere between 2.00-2.75 wide.
2) Most DVDs have 16x9 aspect ratio so you shouldn't be getting any black bars. I haven't watched any newer DVDs for years since I upgraded to blu ray so I might be mistaken about the fact that DVDs do not come with scope aspect ratio. I want to say that's a blu ray thing but somebody will correct me if I am wrong.
3) Youtube videos usually either have 4x3 or 16x9 aspect ratio. I haven't seen many scope ones so it is normal to fill the HDTV screen or have black bars on the left and right. If you do notice a crop between a version on a computer and TV, change the TV's aspect ratio settings which I talked about above.
If you have an older 4x3 TV which I doubt:
1) You will have black bars with most DVD content and even bigger bars with blu ray content if shot in scope.
2) If it fills the screen with youtube videos then you are watching videos there are 4x3 aspect ratio.
I am also thinking that maybe you are comparing the youtube videos on an iPad which almost every time has black bars on top and bottom. Keep in mind that iPad has 4x3 aspect ratio, not 16x9.
