In Act 3, scene 2, how has the play within the play changed Hamlet's situation and influenced the action of the play?

In Shakespeare's, Hamlet, Hamlet
seemingly devises and then reveals his plan to prove or disprove the Ghost's story that
Claudius killed King Hamlet, during his soliloquy in Act 2.2.565-572.  He starts the
speech with angry condemnations of himself, but ends it with the revelation of his
plan.

Hamlet is intelligent enough to know that the Ghost
could be a devil in disguise trying to get him to kill an innocent man, Claudius. 
Though Hamlet certainly doesn't like Claudius, and though Claudius usurped his thrown,
albeit legally, and hastily and incestuously married his mother, Hamlet isn't willing to
kill him without proof.  The plan is to have the players present a play that presents a
murder scene in which the murder is killed in a similar way to how, according to the
Ghost, Claudius killed King Hamlet.  If Claudius overreacts with a strong show of
emotion, Hamlet will know with certainty that Claudius is
guilty. 

Claudius does overreact, of course:  he rises then
yells

Give me
some light.  Away!  (Act
3.2.250)

This is so drastic
that the entire stage clears, except for Hamlet and Horatio.  And Horatio informs Hamlet
that he agrees with his interpretation of Claudius's behavior with "Very well, my Lord
[he perceived it very well]" (Act 3.2.268), and "I did very well note him," two lines
later. 

The confirmation Hamlet receives from "the play
within the play" changes Hamlet's position so drastically that it should end the play in
a matter of minutes.  Hamlet now knows Claudius is guilty, and he sets off to kill him
and get his revenge. 

But, unfortunately for Hamlet and the
others that die at the close of the play, it does not.

In
Act 3.3.73-96 Hamlet has a perfect opportunity--and is now equipped with the necessary
certainty of Claudius's guilt--to get his revenge and kill Claudius.  But he
doesn't.

Why?  Because he mistakenly thinks Claudius is
confessing his sins, and killing him during confession would send him straight to heaven
with his sins forgiven.  And Hamlet won't do that.

And this
is the climax of the play.  When Hamlet refuses to kill Claudius he dooms himself and
Denmark.  Again, why?  That's what happens when human beings play God.  Salvation is
God's business, not Hamlet's.  Hamlet messes where he doesn't belong.  He should go
ahead with his revenge and let God take care of
salvation.

Ironically, of course, Claudius never
confesses.  He, too, is intelligent, and he's smart enough to know he cannot be forgiven
as long as he clings to the rewards he got by committing the
sin.