TL;DR Prue's stubbornness got her killed very early on in the series, something the others never discovered because of her clone taking her place
Prue did not die in the end of season 3; she died as early as in the middle of season one. In season one, there's a episode called "Which Prue Is It, Anyway?", in which Prue clones herself using an incantation in the Book of Shadows, creating two identical clones of herself - this in order to defeat some guy with a crystal sword who wants to kill her for her powers. Anyways, to keep track of who of the three Prues is the 'real' one, Phoebe and Piper decide to color-code them by letting them use different sweaters. The real Prue is supposed to be wearing a black sweater.
After the death of one of the Prue clones, the two remaining Prues decide to confront the evil guy in his own home. Both Piper and Phoebe think this is a really bad idea and tries to talk the real Prue out of it. Prue insists on it being a good idea, as she doesn't want to risk the evil guy finding her first and therefore havníng her sisters being exposed to potential harm. However, as they enter Prue's room, they walk in on Prue and her clone being busy adjusting their clothes. They are both wearing the same outfits as in the scene before, so why would they both be readjusting their clothes like that? My explanation for this is that they changed sweaters with each other in order to be able to go through with the risky plan, as they know Piper and Phoebe will do everything in their power to stop Prue from going. As Prue has forseen, the two other Haliwell sisters demands to talk to the real Prue - the one presumably in the black sweater - in order to stop her from going, as the clones' death have proven to be non-harming to the real Prue. The Prue in the blue sweater is forced to wait outside when they have their talk. When the others are done talking, they notice that the Prue in the blue sweater is missing - she has decided to carry through with the plan.
As Piper and Phoebe has warned her for, the ill-constructed plan fails and the blue sweatered Prue dies at the hands of the evil guy. The Prue in the black sweater feels this but does not disappear - it appears that she really is the real Prue. Here we come to what is my final conclusion - the incantation Prue read from the Book of Shadows created two clones with all of Prue's memories and experiences who would stay as long as they were needed. When the real Prue - the one in the blue sweater - died, she left behind two sisters who not only were very dependant on her but still also possessed fairly weak powers. They would - at the time - be easily defeated by almost any enemy. Because of this, the remaining Prue clone had to stay, as the spell obliged her to stick around for as long as her presence was needed.