The importance of the scene in relation to El Camino is that it's Jesse finding closure with his old (Breaking Bad) life. It's not just Walt, we also see memories of Mike and Jane.
The scene very much highlights a part of Walt's character that defined Jesse's interaction with him, in a way that Jesse now retroactively think he should've seen the red flags.
There are a few small things here: Walt is giving unsollicited fatherly advice, Walt assumes Jesse didn't graduate (it reveals how Walt thinks about Jesse), Walt is generally meddlesome (telling Jesse what to do/achieve) though he intends well. But the most important element here is found in the following quote:
You're really lucky, you know that? You didn't have to wait your whole life to do something special.
It's not the resentment that's the big issue here (though it does contribute), but rather that Walt is clearly in love with what they are doing. Walt's love for his activities is what will eventually lead to his downfall and that of everyone around him, leading to Jesse's life in a cage and subsequent life on the run.
You can even see it in Jesse's response to that quote. He's initially dumbstruck, clearly trying to figure out if Walt meant what he said. Once Jesse realizes that Walt didn't misspeak, he bows his head in avoidance, which implies he disagrees but is not willing to get into it.
While it is only in later seasons that we find out that Walt is in love with building an empire as opposed to making meth or making money; the point of the scene is that Jesse is reminding himself that he should've seen the red flags way back at the diner. He somewhat did notice it then, but he dismissed it and continued working with Walt.
Jesse was seen wearing a cap that Skinny gave him. Is that scene linked to his present?
As @Charles already mentioned, it's not Skinny Pete's cap (which is full black with a white logo). It's Jesse's own cap. It just happens to somewhat resemble it.