I have spent quiet a bit of time hiking and fishing in the park and the bears have never really cause me any concern. I have been extremely close (within 10 feet) to bears in the backcountry, but I have never felt threatened. For the most part the bears I have encountered wanted to go on about their business, and fortunately the bigger the bear the quicker he seems to disappear. With all of that said, I did have an incident 2 or 3 years ago that caused me concern:
I was just below the parking lot at the Abram's Creek trailhead in Cade's Cove. I was in the creek fishing and my wife and young daughter was sitting on a rock by the creek watching. About that time I hear some folks talking about a bear. I am scanning ahead and I see the bear moving toward the stream, with the intent to cross. At this point I have no concern, I am simply hoping he will cross the stream like many have before. But, as he steps into the stream he stops and looks at me. Now I have concern, not for me, but for my wife and daughter. I told my wife to pick up our daughter and move slowly back up the path to the main trail. My concern, the bear was being chased by people with cameras it had intended to cross the stream to get away, I was in the way, now its escape route was down the creek bank between me and the trail above with all the people and now my wife and daughter were in the way. They backed up to the trail and the bear walked down the bank, crossing the rock they were sitting on. The bear passed between us and we weren't 40' apart. The bear went on with all the cameras continueing to give chase. It was all I could do to not yell at someone.
Here is my concern, When people start chasing bears to get their pictures I don't want to be and I certainly don't want my family to be in what is the bears escape route. I never felt threatened. But had my wife and daughter not had a place to move to then that bear was going to have to figure out another way of escape.
When the bears are left alone I don't think they present a problem. Don't worry about them near as much as falling, snakes or bees, unless of course they are being chased by a bunch of people with cameras.