I've never needed a bandage. So, as I said, I have very little clue about bandaging or tourniquets.
Anyway, I'm thinking, tourniquets, stop the blood flow to the limb, and bandages just block the injure, am I right?
A tourniquet is usually made of surgical tubing, basically a big rubber band (though you could improvise one with other materials, a rope could be used, a shoelace, whatever you might have handy), and is wrapped tightly around the limb to constrict blood flow (though as WiFiDi points out, it wont stop blood flow 100%, nor should it).
It is used to stop a patient from bleeding out if they have a wound that's loosing blood rapidly, but as a downside, the patient will loose use of that limb whilst the tourniquet is there, because there won't be enough blood flow to opperate the muscles normally (but there probably woulden't be anyway, if the wound is just leaking the stuff).
Unfortunately though, if the wound is on the torso, it will be quite impossible to use a tourniquet, it only really works if you have a severed major arterie in the arms or legs.
A bandage is usually made of gaze (or could be improvised with any clean cloth you might have handy), and can stop bleeding from something like a flesh wound, and more importantly, prevent further contamination of the wound by covering it up.
It would not be sufficient to stop someone from bleeding out if they have a severed major arterie (though it could buy them some more time if wrapped tightly), but for a torso wound you would have to try anyway, but the patient would be living on borrowed time, and has to get to a surgeon to have a chance.
Bandages are also applied to sprained or broken limbs, having them wrapped will slow down swelling, and more importantly, provide some support and keep the limb in the right place/position, which really helps the pain.
For a broken limb though, you'd have to get a cast on there (made of a solid material, gipsen is the traditional material used, but in recent times, plastic is starting to become very popular), just so you can hold the bone in the right position, so it will grow back togeather correctly.