Where the two differ, of course, is in the context those characters operate in. Shakespeare's characters see themselves at the center of events. Often, this is because they are at the center of things--the histories and most of the tragedies are about political leader, ranging from Hamlet the Danish prince, to Othello the decorated military leader, to Lear the king, to Henry V, hero of English history. And his characters are often subject to interest from ghosts, sorcerers, and other supernatural beings (think of A Midsummer Night's Dream, Macbeth, the Tempest, or even Hamlet, which all feature supernatural elements). These are inherently interesting people in inherently interesting situations.
Meanwhile, Chekhov almost always writes about people who see themselves at the periphery. Three Sisters is a great illustration of this. The Prozorova sisters are living in a kind of exile at their late father's country estate. They are bored with their small town and dislike their dull, unfulfilling jobs. All of them--especially the youngest sister, Irina--pine to return to the Moscow of their youth. But they never make it their. It's a very sad play, but not a tragic one. The sisters aren't undone by some great tragic flaw--they are just stuck.
Chekhov's characters are often stuck. In The Seagull, Nina and Konstantin both aspire to becoming artists--but they never manage to achieve any success at it. Uncle Vanya's title character nearly commits suicide due to unrequited love