To eliminate something means to get rid of it, with the implication that they are removed or moved out of the way. For example: To eliminate an opponent in a basketball playoff game means to get rid of the opposing team from continuing to compete for the championship by beating them in a game.
To eradicate also means to get rid of something, but by destroying it. For example: A pest control service eradicates insect infestations like termites and cockroaches. They get rid of the insects by killing them with pesticides.