There are tons of advantages to each of those phones, and each one has things that make it "better" then the others. All of them are a vast improvement over your S3. Earlier this fall I upgraded my S4 to the G3. Part of it was initial cost - the G3 was on sale where the S5 wasn't. I could have gotten the Note 3 (my carrier didn't have the Note4 yet) but it felt like a the S4, just bigger - and I wasn't convinced I'd be using the S-pen that much anyway. Personally I much prefer LG's skin over Android then Samsung's - but then again if you're used to the S3, its a very easy transition to a newer Samsung.
I really like my G3, it's a much better phone (in my opinion) then the S4, but I'm sure there are people who would argue that. For me and my usage it's pretty much perfect. There are only 2 things I miss or wish I could change. I like the screen size (5.5 inch) but I prefer Samsung's AMOLED screen technology over LG's LCD. Not saying either one is perfect, but I really like the slightly over-saturated colors and almost perfect blacks that you get with AMOLED. The other thing is build quality. I find Samsung has done an incredible job with their battery door and it's latch system. I haven't removed the battery door from either my S4 or my G3 very much (and probably even less on my G3 compared to the S4). However the G3 developed creaks and slight flex in my hand that I never experienced to nearly the same degree on my S4. Overall the plastics and shape feel far better on the G3 (not as slippery, easier to hold), but the S4 didn't develop the creaky, flexy feeling that my G3 has.
Camera is again a matter of taste. The S4 tends to oversaturate the pictures and make them slightly brighter overall then the actual scene. The G3 is slightly undersaturated (colors are more muted and dull) but the exposure seems to be more accurate and even. Also, the G3 camera has optical image stabilization that the S4 doesn't - BUT the S4's stock camera app has far more options, settings and control over your shots then the G3. The G3 is very much a point and shoot with little control over the image settings.
Size - none of these are exactly small devices, but the Note3 felt really awkward to even attempt to one-hand anything. Partly due to shape, partly due to the larger size (and larger bezels compared to the G3).
Now, I realize that I compared a lot to the S4, but it's the phone I have experience with that's closest to the ones you listed. The S5 and the Note 4 are both improved over the S4. Hopefully some of this info helps. Also, the newer the device, the better chance of future updates. Not everyone cares about it but it is something to think about.