The vocal cords or vocal folds are necessary for speech, and we generally think of this as their prime function. This might even be their prime function, as they have many more fancy design features to maximize speech production than for any other use. But other functions they do have, some of them quite important.
The vocal cords can clamp shut very tightly, and do so automatically if anything coming down the windpipe touches them. This, together with the choking and coughing reflexes, have saved you from some very nasty lung infections, and has probably saved your life. Every time you 'swallowed something the wrong way', whether it was a sip of water or a splinter of bone, you were at risk of dying. Even when you are coughing and red in the face, chances are that whatever you swallowed didn't make it past the vocal folds.
Somewhat less dramatically, the vocal cords clamping down will also trap air in the lungs. This allows you to hold your breath, which can be quite useful, but it also provides a structural function. Filling your lungs with air and tightly shutting your vocal cords will 'brace' your thorax, providing a stiff framework for your muscles to work off of. This is called thoracic fixation, and we use it unconsciously, but constantly.
Perhaps the most obvious example of thoracic fixation is seen when you are lifting a heavy weight, and you grunt. This grunt pops out as your body braces itself by clamping down the vocal chords. You may also find yourself gasping or giving a deep sigh when dropping a heavy weight, as your body relaxes after a hard fixation.
Another good example of thoracic fixation is giving birth. Bracing the thorax gives the abdominals something to push from, and will make the baby come out faster. This is why you might hear the expectant mother being told to "breathe!" No, she isn't going to suffocate without the doctor's instructions. Concentrating on taking deep breaths will help the mother to keep from pushing with the contractions until it's time for the baby to come out.
And now you know why the rest of the animal kingdom has vocal cords.