The terminology we use in martial arts is basically what Ganryu Kojiro outlined earlier: strangle = cut off blood supply to the brain; choke = cut off air.
Forensic terminology is not quite as exact about the mechanism, I believe, since pathologists are generally dealing with people who aren’t as precise and knowledgeable about what they’re doing. In forensics, strangulation is used to mean both cutting off blood supply and air, while they make a distinction about how the pressure is applied: manually (by hand, with no artificial aids) or through ligature or hanging.
Movies almost never show either how long is actually needed to kill someone from strangling, nor how violent a life and death struggle is. Naturally, they’re more concerned with story-telling than in showing realistically all the aspects of the killing.
The short answer to how long to die is: about 10–15 minutes, but you’ll get unconsiousness in as little as 10–15 seconds.
If you wanted to show realistic action, the movie killer should primarily cut off the blood supply, and only secondarily think about the airway. To to be sure the victim is dead, you’d need to keep pressure on for at least 3–5 minutes with a good (martial arts terminology) strangle that more or less completely shuts off the blood supply to the brain. This is enough time to cause cessation of pulse and respiration, and is around the time needed for brain death.
If you’re talking about strangling with a cord or rope, or by manual strangulation, your killer should make sure pressure on the victim’s neck is maintained for for 10–15 minutes after unconsciousness. Less time than that is not going to be 100% sure to kill someone.
If the airway wasn’t crushed, it’s possible for someone who looked “dead” to come back from being strangled. Unless the killer held the pressure on for several minutes, it’s unlikely that the person would actually be dead, though as earlier noted only about 3 minutes of severe oxygen deprivation is enough to cause brain damage, even if the person doesn’t actually die from it.
With strangulation using a ligature, or in a hanging, it’s an either/or/both mechanism, depending on where the rope or cord ends up, where the knot is, how it’s tied or pulled, how far the person falls, how much pressure there is being applied, etc. As Gfactor cited, short drop or suspension hangings will probably (though not always) produce fairly quick unconsciousness from arterial compression, but actual death from asphyxiation could take a very long time.
In fighting, strangles get fast results. You can put someone out in seconds if you’re right on target, almost certainly in under a minute. Even when we were being careful and considerate of our partners in the dojo, and not really cranking them on, we’d sometimes get narrowed vision and haziness after only about 5–10 seconds. Strangles are very effective if properly applied, and are not usually lethal or even particularly damaging, since you’re not tearing or crushing tissue.
The problem with using them for self defense is that very occasionally you’ll get some nasty feedback from stimulating the vagus nerve and depress or stop cardiac activity. If you don’t actually want to kill the person, this could be quite awkward. In the question the OP is asking, you shouldn’t count on this to produce a dead body.
Chokes (martial arts terminology again) involve closing or crushing the airway. You will always cause more trauma with a choke than a strangle. The way the trachea is built, if you crush the cartilage that supports the airway it will close down and stay closed. (This is when you’d need to do an emergency tracheostomy if you don’t actually want the person to die.) It’s also possible to crush or rupture the larynx, or break the hyoid during a choke. This is actually very likely with a pure choke since the person will be conscious and struggling for some time.
If you’re trying to stop a nasty fight quickly, wrapping a cord around the neck will compress both arteries and airways, putting him/her down pretty quickly, while simply choking across the throat with no wrap will do nothing much to cut off blood supply and the person will be conscious and extremely active and agitated for quite a long time, which is obviously less than ideal (in the clawing, kicking, biting, bucking, throwing you around sense of that term).