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 <I>mostly </I>the same person.
<P>
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.
<P> 
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.
<P> 
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.