A tame animal is one that is relatively tolerant of humans. A wild animal can be tame if it has a lot of positive exposure to humans, although it is still a wild animal with all of it's instincts and behaviors intact. Unfortunately there are many examples of tame animals turning on their handlers/owners -- from Roy Horn (Siegfried & Roy) being attacked by his tiger during a performance to the woman who was attacked by a friend's pet chimpanzee and had her face and hands ripped off.
A domesticated animal is different than a tame animal. Domestication is a process whereby humans genetically change an organism to meet their needs. When an animal is domesticated it loses many of it's natural behaviors and instincts, and tends to be able to be tamed much easier when socialized from a young age.
For years I have opposed traditional circuses that use tamed, wild animals to entertain audiences through trained performances. A large part of my opposition is due to the fact it has been documented that many of these animals are trained using highly aversive and abusive methods. These methods are unnecessary -- through my own education I know that it's possible to train any animal through the use of positive methods.
A bigger concern is that circus animals cannot have the proper environment and enrichment to thrive and engage in their natural behaviors. When traveling from location to location animals are transported on a train. In between performances on the road elephants are chained by shackles on their feet. Wild cats are kept in barred cages much smaller than even the habitats that good zoos use to house their big cats -- which is smaller than what their habitat would in the wild.
Humans share the earth with many other species. As the dominant species we have a responsibility to care for the earth and make choices that not only benefit us, but that treat animals in a humane manner. Using animals solely for the purpose of entertainment when it doesn't benefit the individual animal nor the rest of their species is, in my opinion, immoral. Additionally it sends the wrong message to children - that wild animals should be something to be entertained by, and not respected.
Recently Ringling Bros. recently announced that they are eliminating their elephant acts, and will phase them out by the year 2018. Although too long in coming, this is a bit of welcome news. Now Ringling just needs to commit to eliminate all of their wild animal acts.