First of all, if you have been, or suspect you may have been - bitten by a spider; seek medical help as soon as possible.
That said, basically creates an awareness that any foreign substance: be it venom, virus, germ, dirt, disease etc. in the body; can lead to death if not taken care of properly.
Most spider bites have resulted in no deaths in our contemporary advanced world, where hospitals and doctors are available in nearby locations. This doesn't mean that an untreated spider bite wont kill you, specially if you're allergic to venom. Spider bite deaths have been relatively uncommon, due to those who have been bitten, taken the proper measures.
This question may have different answers, providing we take into consideration the age of the bitten victim. Of course, special immediate care should be taken in all cases; and most importantly in the young and the elderly victims. Spider venom is very slow acting in some species, and bites may begin to show pain or itching hours (1-2 hours) and even days (up to 36 hours) after where the patient may be, (or not) aware of the bite. In untreated patients these spiders may be able to kill humans if bitten: