Star Wars: The Clone Wars bridges the gap between Attack of the Clones and Revenge of the Sith and chronicles the events of the titular Clone Wars where the Jedi and the Republic face off against the Separatists. Season 1, Episode 1, "Ambush" follows Jedi Master Yoda on a diplomatic mission which is derailed by the sinister Count Dooku and his Sith apprentice, the assassin Ventress.
During Dooku's attack on Yoda's fleet, the Jedi Master makes his escape, along with three Clone Troopers, via an escape pod. He is warned that the Separatists will shoot any escape pods leaving a ship, so he deploys all of the escape pods as a distraction. His plan is successful despite the fact the Separatists do indeed shoot at the escape pods. This scenario demonstrates just how pivotal the Empire's decision to not destroy C-3PO and R2-D2's escape pod in A New Hope was.
At the beginning of A New Hope, Princess Leia and her band of Rebels are on the run from Darth Vader having just successfully stolen the plans to the Empire's Death Star base. She puts the plans inside R2-D2 along with a message to Obi-Wan Kenobi, pleading with him to aid the rebellion. She is ultimately captured by Darth Vader, but C-3PO and R2-D2 successfully escape via an escape pod. When their pod jettisons from the ship, an Imperial Officer wants to shoot it; however, he is stopped by another, clearly superior officer who says not to bother since the pod shows no signs of organic life.
This decision to allow the pod to escape is foolish for numerous reasons. The first is that droids are very common in the Star Wars universe and not a Rebellion novelty. Those two officers not only know that droids exist, but understand they can function as spies and Rebels the same as organic life. Even if they don't assign intelligence or understand droids to have personality, they would know that droids could be programmed by Rebels to collect and return information independent of any semblance of free will. Also, what if the Rebels had simply put the plans on the pod alone?
Additionally, blasters and canons don't seem to have a limited ammunition. It costs the Empire nothing in terms of supplies to destroy the escape pod with a canon. It wouldn't be wasteful of resources to destroy the little vehicle even if it wasn't carrying the Death Star plans. This fact is why it's best to destroy all escape pods as demonstrated in The Clone Wars. As mentioned above, Yoda barely escapes capture or death thanks to this very strategy, but the Empire apparently doesn't continue it beyond the end of the war for an unclear reason.
The scene in The Clone Wars just highlights how ridiculous it is that C-3PO and R2-D2 escaped Darth Vader in A New Hope. The failure to destroy that escape pod not only directly leads to the destruction of the Death Star, but to the Empire as a whole, which makes the decision even more ludicrous.