This case seems so obvious, but nevertheless has much to tell us. The obvious answer is yes: the old woman and young girl are the same person. A little deeper thought, however, may real a more accurate answer: the old woman and the young girl are mostly the same person.
A bit of confusion can result from terminology if care is not taken. We use the word "same" in what appears to be two ways. First, when we say "I'm not the same as I was yesterday," we are indicating that some change has occurred, i.e. there are some properties which differ between yesterday and today. We may nevertheless say, "I'm the same person I was yesterday," meaning that some central thing -- our identity -- remains the same despite any changes in outward properties. This somewhat confusing (if not contradictory) position results directly from a Boolean notion of identity.
With a fuzzy perspective, however, the difficulties (and the two meanings of "same") vanish. To speak more precisely, we would say, "I'm not quite the same as I was yesterday," but if pressed, we'd have to admit, "I'm mostly the same person as I used to be." These two statements are not contradictory, because of their fuzzy nature. One might be fairly tall for a fighter pilot, but rather short for a basketball player. In the same way, any changes in me since yesterday may be enough to make me remark on the difference, but still very minor in comparison to the differences between individuals.
Returning to the old woman, we can clearly see that she is not exactly the same person as the girl who scraped her knee. This woman has skills (driving a car), knowledge (of President Clinton and microwave ovens), memories (of her wedding) and personality traits (perhaps a bit more caution and prudence) which were not possessed by the girl. If they could be put in identical situations, they would react in different ways. Yet the two share a great deal, as well; many of the girl's memories and other mental traits are still in the woman. They share more, in fact, than any two people which could be found to exist in different places at the same time (assuming no futuristic technology). Thus, in the 20th century and before, it is sensible to call all the stages of a person's life the "same" person, and someone who shares no memories a "different" person. Strange cases which may occur in the future will cause this convenient division to break down, and we will need to speak more accurately: the old woman is mostly -- though not completely -- the same person as the young woman.