The main difference between the two stories is their central conflict.
"The Most Dangerous Game" is a classic Man vs. Man story. Rainsford is hunted by General Zaroff, and he must survive. The two characters are similar in their skills and experience, but different in their motivations. Rainsford fights to survie, while Zaroff fights for the thrill; their personal conflict is what drives the story forward.
"Leiningen Versus the Ants" is a classic Man vs. Nature story. Leiningen needs to fight off the advancing horde of army ants with his own skills and knowledge. The ants have no intelligence driving them; they are simply a force of nature willing to destroy themselves to move forward. Leiningen uses his selflessness and reasoning ability to win, at the cost of replaceable crops. Leiningen fights to protect his plantation and men, while the ants fight because that is their instinct.