So, I wonder if the G3 and S5 screens may show a similar difference.
They aren't. You saw a difference between your S5 and Note 3 because the Note 3 has a larger display, the same resolution, and you had to get really close. Instead it would be like comparing the S5's display to the Note 4's when getting that close. However, like you said, you had to get really close. The differences are not observable at a normal viewing distance.
The ppis are higher for a reason.
Yes they are and it's all a numbers game. Look, you can state your opinion all you want and that is fine but there is scientific reasoning behind what I am saying. It is not a matter of opinion but a matter of fact. Companies are cramming more and more pixels into their displays because it's all a ploy. Higher specs automatically mean a better experience, right? Go back and watch the LG G3 commercials. They aren't comparing their phone to the S5 or any other Android smartphone but rather the iPhone. Why? Because the iPhone is their target and they only care if their phone has better specs than it. I suggest you read up on this article to get a better understanding of what I'm talking about: AnandTech | The Pixel Density Race and its Technical Merits
Again, I'm just looking at the fact. It is fine if you think the G3's screen looks superior but the perceived "sharpness" that you are seeing is a result of the placebo affect (your brain knows it's higher so it must be better). At a normal viewing distance, there isn't a difference in sharpness between the S5, Note 4, G3, or even the M8, Moto X, or Nexus 6.