@MikeMason-- It may be necessary, especially for children. It's also necessary if the bite becomes infected or doesn't resolve within a few days.
I was given a tetanus shot as well as antibiotics when I got bit by a wolf spider bite. My bite was infected and basically turned into an ulcer. But the doctor at the hospital said that not everyone develops an infection after a spider bite.
So you should look out for signs of infection. If the bite is very swollen and inflamed, if it becomes ulcerated and starts oozing pus, you need medical attention.