Rival, compete, vie, emulate can all mean to strive to equal or surpass another or his achievements.
Rival (see also MATCH ) usually suggests an attempt to outdo each other.
Compete and vie, usually with the opponent explicitly stated after with and the objective after for, may sometimes omit direct reference to one or both of these.
Compete stresses a struggle for an objective (as position, favor, profit, or a prize); unlike rival, it need not suggest a conscious attempt to outdo another but may imply a quite impersonal striving.
Vie carries less suggestion of arduous struggle to hold one’s own or to excel than compete, but it may suggest more conscious awareness of the opponent.
It sometimes suggests the excitement of contest that is a game rather than a combat.
Emulate implies a conscious effort to equal or surpass someone or something by imitation or by using him or it as a model.