Why didn't Aragorn Let Theoden kill Grima?

There's a recurring theme in The Lord Of The Rings, both novels and movie, that showing mercy towards wrongdoers is an act of highest morality, and that killing should be done only for defense, not for revenge.

The clearest example of this is Gollum. In the first movie there's this bit of dialogue, taken verbatim from the books:

                    FRODO
                    (grim)
         It's a pity Bilbo didn't kill him when he
         had the chance.

                   GANDALF
         Pity? It was pity that stayed Bilbo's
         hand. Many that live deserve death, and
         some that die deserve life. Can you give
         it to them, Frodo? 

Frodo frowns.

                   GANDALF (CONT'D)
         Do not be too eager to deal out death in
         judgment...even the very wise cannot see
         all ends. My heart tells me that Gollum
         has some part to play yet, for good or
         ill, before this is over. The pity of
         Bilbo may rule the fate of many.

This sentiment by Gandalf is, of course, foreshadowing Gollum's importance later on, and his essential role in destroying the ring.

Aragorn, in this case, is showing the same mercy that he doubtless learned from Gandalf. And not surprisingly, the same mercy does, later, come back to help the good guys, to Saruman's downfall - in the books he is responsible for Aragorn taking possession of the Palantir and later on for Saruman's death during the Scouring of the Shire. In the movie, we have this scene in the Extended Edition of the Return of the King where Grima redeems himself:

Grima Wormtongue appears behind Saruman. 

                              THEODEN    
               Grima! You need not follow him! You were 
               not always as you are now. You were once 
               a man of Rohan. Come down. 

Grima looks like he going to come down. 

[..]

                              THEODEN
               Grima…. Come down! Be free of him! 

                              SARUMAN   
               Free? He will never be free!

                              GRIMA 
               No!  

 Saruman turns around to Grima! 

                              SARUMAN
               Get down cur! (slaps him to the ground) 

                              GRIMA
               Ah! 

[..]

Grima rises up behind Saruman and attacks him with the knife, 
stabbing him twice in the back. Legolas aims an arrow at Grima 
who falls shot in the heart. Saruman falls from the tower and is 
impaled on a great wheel below.