It certainly does make her act differently, for some very different reasons. First, she feels an obligation to be strong, to conceal her problems and losses for the sake of her mother and sister. She loves them, and she doesn't want them to worry about her. Second, she puts on a strong front for the powers that be, to be defiant in her own way, and also, to discourage them from manipulating the game in a way that would make things worse for her. Finally, Katliss has enough vanity in her for her to want her audience to admire her. And really, would any of us be "ourselves" on live television, even in situations less consequential than the hunger games? The fact that she does act differently and her reasons for doing so make her a more realistic and well-rounded character, which enables us to identify more with her and care more about her.
Hard to tell because there are other factors that contribute to how she acts such as the fact that there are 20 other kids trying to kill her at the same time that and the fact she is away from home for a long amount of time but if you put all the factors together the fact that it is televised is like icing on the cake.