Character mistake: After distracting the guards, Danno tosses a grappling hook onto a ledge and climbs the side of the building. He then goes about his business, carelessly leaving the hook and rope dangling from the ledge for all to see. For a seasoned 5-0 team member, that's pretty sloppy covert operations.
Character mistake: Cameron's attempt to assassinate the governor fails because McGarrett has everyone wear bullet-proof vests. It's been established that Cameron is brilliant, a top-of-his-class ex-cop and a crack shot. And he's been sending death threats, so he knows they'll expect him. Yet he fires at the governor's back and is shocked that the man doesn't die. Anyone with his expertise would have anticipated the vests, and aimed for the man's head.
Character mistake: When Ben talks to the old guy about Spider Brown, Ben says, "Somebody that got rich the past week." The time between the shooting and trial was probably months, not just one week. Spider would have been flashing money right after the hit, and then disappeared.
Character mistake: McGarrett tells Pereno that the sabotaged race car went over a cliff. But the car crashed into a wall, with no cliff in sight. It's very unlike Hawaii's top cop to get the facts wrong - especially when he was an eye witness to the event.