My impression was they took the Rebels more seriously. So sent the Emperor’s blood hound after them with a small fleet and a fancy command ship. Palpatine is playing with fire knowing that if Vader finds his child there will be sparks. But he also knows it will be a beacon to finding the rest of the Rebels that fled Yavin.
I sort of get the impression that the Emperor lost faith in guys like Tarkin after the first Death Star fiasco, and gave Vader more autonomy.
I attribute it to the character and film being a global cultural phenomenon lasting years, so Lucas decided to change his significance and history to increase profits. No story beyond '77 suddenly became hire someone to write some sequels, this thing is huge.
I think Lucas exaggerated how much of the early story he shaved off to make the first film. It is the first third, tightened up, with a bit of the middle and a scaled back space battle. Hoth in TESB and Tatooine in ROTJ had to be added. The imperial prison planet became Bespin, the Luke Vader duel was split, and Ben training Luke became Yoda training Luke. A hell of a lot of story got added with considerable duplication. That continued with the PT reusing names and more from early drafts.
From what I read, Vader fell victim to his own popularity and the ANH costumes weren’t up to a sequel. And while Vader is in charge in TESB, he isn’t in ROTJ. Vader is Tarkin’s lackey in the first one and the Emperor’s in the other two. He really isn’t portrayed as the badass he is in popular culture. We are always reminded he has a boss he has to answer to. Just in TESB it is subtle.