I think that Othello's characterization is enduring because it strikes at the heart of what lies at our most elemental fears of our interaction with other people. Essentially, Shakespeare creates a character that is paralyzed by the fear of whether or not others see him as he sees himself. Othello recognizes, painfully at times, how much of an outsider he really is. He is an outsider as a warrior, and distinct from the landed political power setting into which he is thrust. He is not a descendant of established wealth, and he is a man of color in a world where this is not the norm. Adding to this is the fact that he cannot fully accept the fact that Desdemona really does love him for who he is. All of these conspire to plant the seeds of doubt and insecurity within him, reflecting the fundamental fear of being seen by others as he sees himself in his darkest of moments. In doing so, Othello's characterization is lasting and powerful because it is a reflection of who we are at our very worst. The casting of aspersions of another because of our own doubt, or wondering at what point someone will leave us, or doubting the very basis and fabric of our own relationships are uniquely human characteristics, and also represent the basis of Othello's characterization, making them and him resonate and reverberate in our minds as an example of how bad we can get in our desire to be loved and accepted by others. It is for this reason that what Shakespeare develops is an enduring portrait of both his title character and ourselves.
If you read critics of Othello like A.C Bradley and Samuel Taylor Coleridge, you'll often find that their greatest appreciation of Othello lies in the characterisation. A.C Bradley describes it as "the spectacle of noble beings caught in toils from which there is no escape". This is exactly what I feel; unlike Shakespeare's more 'grand' tragedies about aging kings and events of generally epic proportions, Othello is all about these characters with the most amazing 'spirit' of moral integrity caught in this unescapable web woven by this thouroughly evil master manipulator. The character of Iago is another genius of Shakespeare-he's just so evil, and so intelligent, and the funny thing is although he gives all these reasons for why he hates Othello, you get the impression that he really does it just because he can and he has to make up these justifications to convince himself he has a reason; an Elizabethan pshychopath you could say!