My buddy wayne seltzer summed up the science stuff nicely. So I'll drop that here first....
The differences between the 1.8 and 1.4e are:
1) 1.8 has much weaker correction for SA and thus will have focus shift which will throw off the AF if you take pictures stopped down to say 5.6 and the subject is close enough to not have a large enough DOF to hide the focusing error.
2) The 1.4e is much sharper in the 1.8 to 2.8 aperture range, especially in the center.
3) The 1.4e has much better coma correction in the corners which will lead to better AF on objects towards the edges of the frame and better for astrophotography.
4) The 1.4e has more ED glass and thus has less LoCA than the 1.8, although it is far from APO itself.
5) The 1.4e has better flare/ghosting handling than 1.8.
You obviously pay more for those extra ED elements and aspherical elements which give you the better performance mentioned above. Whether you see the difference is dependent on the type of shot and whether that extra performance is worth it to you.
I don't think the 1.8 has ANY ED glass in it, though, so one correction for wayne on the above.
And I have this to add, myself:
The 1.8G is a tricky lens. It produces smooth images and is a relative steal.
I do agree, the focus shift on the 1.8G was very hard for me to figure out, when I owned it. I owned it twice, too. Actually, I don't know that it can be tamed.
On the other hand the 1.4E doesn't have any odd curve at all. In fact, it is easily the most enjoyable 1.2 or 1.4 lens I've ever used or owned, with regard to focus shift and shooting wide open.
And I've had the following: 24 1.4L II, 35 1.4L, 35 1.4L II, 35 1.4 sigma art, 35 1.4G, 50 1.2L, 58 1.4G, 85 1.2L II, 85 1.4G, 105 1.4E. The 105 1.4E easily comes in second for easiest 1.2 or 1.4 lens to shoot wide open. So it must be in the new technology that Nikon is using.
Anyone intrigued by the 28mm 1.4E should definitely start with the 1.8G for a few months. Just to make sure he/she likes the focal length. I've said that before, but it's worth repeating... people buy a lens, and soon say "I can't get used to this FL". And 28mm is one of those. 24/28/35/50/85, love it or hate it! But I'd say 24/28 are the toughest for folks.
I think for me, I love the ability to shoot wide open, so comfortably. No curve to manage. Wide open & SHARP! But I also love the versatility. I have a 2 year old. So while I prefer to just use a backup lens, like my 50mm 1.8G when I'm shooting at home, there are times when I am indoors and want something wider. 35 is not wide enough for indoors, for me. The 28mm is nice here. For shooting groups, professionally, the 28mm is a better choice there. It came in handy at a wedding Friday night. We were stuck on a small dock, and the 35 wouldn't have worked, literally. We would have been forced to use a 24-70.
Size/Weight wise, the 28E is just a tick longer and heavier than a 35 1.4G. And a little longer than the 28G, and a LOT heavier. Except when you're weighing the cost.
Finally, the 28E is just flat out the best lens in it's class. It's ED and then some, it's FL. It's going to be better on a D800/810 body with all those megapixels. And so it's future proof and ready for the D850.
If you're on a budget, the 35mm 1.4G at $950 is an amazing professional lens. The 28mm 1.8G is an amazing wide angle for the money. If I were a pro, I'd get the 35 1.4G if money were tight, and skip the 28mm 1.8G, but that's just my opinion.
The E lenses are a HUGE success. The 105, the 28, the 70-200. I've had them all. They are without a doubt, superior to their peers. Nikon's new slogan should be, you gotta pay to play. Because, the new E glass is without a doubt pricey. If you feel that they are not "better" - you are just either not taking enough time to recognize and appreciate it. Or you're just thinking with your wallet, which is absolutely fine!!
With all of that said, if you're asking if it's worth it, it's probably not. $1,995 USD is crazy money, to spend on a wide angle prime. But you know me, C R A Z Y.
p.s. This and the 70-200E, and 105E, are three lenses that required either ZERO AF fine tune or up to +3 on all three of my FF bodies, d810 d750x2. A 35 1.4, 58 1.4, 28 1.8 24 1.8, these lenses were at north of +8, up to +16.