How might Hester's life have turned out differently if Pearl had not been born?In Chapter 13, this might be explained, but I have a hard time...

Assuming everything else, including her relationship with
the baby's father, were the same, the one exception being the birth of Pearl, it is
likely that Hester's life would have turned out much
differently.

Chapter 13 focuses on the changes that have
taken place in Hester in the first 7 years of wearing her punishment.  These chapters
tell of Hester's goodness and the way she uses her talent with the needle to provide for
the poor.  In the eyes of many, the "A" on her breast comes to stand for "Able" rather
than "Adultery."  As a result of wearing this Scarlet Letter, however, Hester has grown
old and wise beyond her years.  While the letter has robbed her of her beauty, it has
given her a sense of compassion and understanding she might never have otherwise
discovered.

Ironically, Hester does not view her new self
with a sense of pride nor contentment.  She longs to feel equal to the other women. 
Knowing this is never possible, she continues in silent misery, accepting the punishment
for her actions, but refusing to be forced into a sense of
guilt.

I think if Hester had never had the baby, and the
public never knew of her sin, any number of things would be different.  Because she was
so beautiful and so talented in creating things of beauty, it is possible the women in
the colony would have hated her anyway.  It is also possible that the secrecy of her sin
would have begun to wear at her inwardly in the way it wears on
Dimmesdale.

There are almost an infinite number of
speculations one could make about how Hester's life might have
been, but I think the purpose of Chapter 13 is to show some of the positive
things that resulted from Hester's mistake.  Perhaps in other circumstances, she never
would have gained the sense of selflessness and charity she develops.  Certainly she
would not have received the almost 6th sense she has of empathy and
wisdom.