Overall the Extended Cut has problems with contrast. There are so many scenes where you can spot black crush or overblown highlights. Part 1 (The Fellowship of the Ring - Extended) has this horrible green tint. Part 2 (The Two Towers - Extended) has the green tint removed for the most part, but still pushes green a little bit more than the Theatrical Cut. Part 3 (The Return of the King - Extended) has basically the same color grading as the Theatrical Cut.
The main benefit of the Extended Cut:
It is the sharpest and more detailed release, with a better encoding and less DNR applied. There are still some DNR-scenes but they exist in all released versions.
The Theatrical Cut is also a mixed bag. Out of the older Blurays it has by far the best and most consistent color grading. But Part 1 (The Fellowship of the Ring) is a blurry mess, full of DNR. Part 2 and 3 look much sharper and almost reach the level of the extended cut (the difference might be due to the different encodings).
The 4k-Bluray-Version made huge changes to the color grading. Some people love it, others hate it. But for me it has the best color grading and contrasts of all releases. The main problem of course is with the overall usage of DNR.
Some scenes are filtered to death and look horrible and worse than on the older Bluray-Releases. All the scenes with CGI-Characters (Gollum, Fellbeasts of the Nazgul or the Mûmakil) look fake to me. In the older Blurays Peter Jackson decided to apply “Fake-Grain” onto these CGI-Elements to make them look more real and imperfect and it worked quiet well. For the 4k-Version they decided to remove this added Grain again. Now the CGI-Elements stick out like a sore thumb. The added Grain at least gave Gollum and Co some texture. The Rest of the movie looks okay, but I could not spot any noticeable upgrade in terms of sharpness and details.
I would say:
Using the Theatrical Cut is the best overall choice for an 4k-Remaster, even though the first movie had not much detail in it to take advantage of the upscaling.