In the attack on the Jawas on Tatooine in A New Hope the Stormtroopers went to some lengths to indicate to observers that Sandpeople carried out the attack. They may not have fooled Obi-Wan but presumably others would have believed that Sandpeople were responsible. What is beyond doubt is the cover-up attempt.
Luke: It looks like the Sandpeople did this. There's Bantha tracks. They've never hit anything this big.
Obi-Wan: They didn't. But we are meant to think they did. These tracks are side-by-side. Sandpeople always ride single file to hide their numbers.
Luke: These are the same jawas that sold us R2 and C-3PO.
Obi-Wan: And these blast points, too accurate for Sandpeople. Only Imperial Stormtroopers are so precise.
What did the Empire hope to gain by covering up the attack? Would it really matter if people knew that they'd done it? They were, after all, in a pretty strong position at this point. The Jawas themselves hardly pose much of a threat. Even if the population of Tatooine was unhappy by the attack what did the Empire have to fear from that discontent. Especially in the light of this recent development.
Tarkin: The Imperial Senate will no longer be of any concern to us. I've just received word that the Emperor has dissolved the council permanently. The last remnants of the Old Republic have been swept away...The regional governors now have direct control over their territories. Fear will keep the local systems in line. Fear of this battle station.
Why not just be open about the brutality of the Empire and their methods?
Edit: Whilst the linked question is along the same lines, neither answer deals with the Tarkin statement. The accepted answer argues that the attack on Tantive IV/the Jawas would cause unease in the Senate. This is rendered irrelevant on the disillusion of the Senate. The other answer says that killing Jawas is bad publicity for the Empire. As I say in the comments, if the Empire cared about this then they wouldn't have destroyed Alderaan. I'd like an answer that preferably deals with the Tarkin comments and the Empire's fear-based strategy. I've changed the question title accordingly.