Okonkwo is different than western heroes because there is no one to challenge directly. The cowboy challenged the rustlers and someone may have died in the shootout. Regardless of whom might have died, the challenge was seen as a righteous one.
Okonkwo's challenge to the guard left the guard without a head and Okonkwo without anyone believing his cause was noble or righteous. His impulsiveness led him to his position of death, which he chose to impart upon himself rather than anyone who was outside the tribe. He believed he had honor, at least, from carrying out his own death even if he had none in the challenge to the newly established order.
The Western hero is strong, handsome and sensitive. He is able to "leap tall buildings" and handle any crisis, but is also capable of feeling and showing love. Indeed, it is his love and his compassion for others that often leads him to take on the dangerous obstacles that solidify his hero status.
Trapped by the biases of his tribe and his culture, Okonkwo hides all feeling of sympathy and compassion and is thus distinguished from Western heroes. He is strong, larger than life, able to conquer obstacles. However, he is haunted by his effeminate father, terrified that the tribe will connect him with the personality traits that are so reviled in the community. Therefore, Okonkwo overcompensates. He rules with an iron fist, shows little mercy and no compassion. In Western literature, he would be best cast as the villian. In this novel, his successes make him a [tragic] hero.