Eckels in "A Sound of Thunder" is shown to be a man who has bitten off more than he can realistically chew. As he sees the sign advertising the Time Safari at the beginning of the story, a "warm phlegm" gathers in his throat, and as he is offered a chance to tear up the check, his fingers twitch, showing his nerves and foreshadowing his failure to face the T-Rex. It is when he does confront the T-Rex that Eckels completely despairs of getting out of this experience alive, which leads him to leave the path and crush the butterfly which is enough to change the future.
Travis, in comparison, is presented as the guide of the group, confident and knowledgeable about how the time machine operates and the various rules that need to be followed. It is he who is entrusted with ensuring that what they do in the past does not influence the future, and thus he is a very angry at what Eckels did and how he endangered the future, threatening to live him there in the past and forcing him to get the bullets out of the T-Rex. He is a strong man, in comparison with Eckels, who knows what is needed to be done and acts upon it.
Interestingly, when we compare both of these characters, it is Travis more than Eckels I think who appears as the hero of the piece. Eckels is presented as a weak man who is easily overwhelmed by the T-Rex, in spite of his hunting experience. It is Travis who shows courage, strength and resilience, even if we might question his treatment of Eckels and the final action that leads to the "sound of thunder" that closes the story.