There are too many shared themes to be coincidence. I suspect that Donaldson consciously used a lot of Tolkien's concepts to allow him to focus on what he wanted to explore. If you lean on some existing material, you don't need to focus your energy on building those pieces. I think Donaldson wanted to explore somber adult concepts about life, illness, mental stability.
Lord of the Rings is high-quality fantasy and provides a rich world of its own that you can disappear into if you like, but it has very limited application to our lives. Thomas Covenant is more likely to mean something to you in your life aside from entertainment. It won't for some people, but for others, it definitely will.
Lots of similarities, which in some cases are cultural touchstones that long predate Lord of the Rings and in other cases seem difficult to explain other than by Donaldson's interest in Tolkien ...
- A ring that contains power. That power can be used perfectly by those with great knowledge or their own intrinsic power. It is useless or worse for those without knowledge.
- An evil lord who wants the ring for his own purposes, and focuses his efforts on acquiring it.
- That evil lord is in rebellion of some sort against the real "god" or "gods" of the books. Those gods, however, are constrained not to directly meddle in the world.
- If the evil lord gets the ring, his powers will grow enormously - allowing him to conquer the world (LOTR) or destroy it and attack the Creator (Covenant).
- Magical wood folk in one, elves in the other. Magical stoneworkers in one, dwarves in the other. Ents in one, forestals in the other. Orcs in one, cavewights in the other. Nazgul in one, ravers in the other. Isengard destroyed by ents in one, Lord Foul's army destroyed by the forestal in the other. Wizards (each with a staff) in one, lords (each with a staff) in the other. Lost Elven magic (Third Age vs. Second Age) in one, lost wards of lore in the other.
- When the evil lord loses, he doesn't typically die, he typically loses his power and is forced to spend hundreds or thousands of years recovering it. Ultimately, he's put in a state of eternal impotence.
I think it's difficult to overstate the LOTR influence on Donaldson.
That said, the themes of the book are far darker in a fundamental way. We are asked to sympathize with Frodo's struggles as he wrestles with the ring's growing influence on him, and we're asked to accept that if Sauron wins, things will be really bad (presumably because he'd be a tyrant), but neither of those concepts really is that strong unless we buy into it.
But we understand Covenant's struggles with himself. They are vivid and powerful. And, we understand (especially in the second trilogy) that Lord Foul is really making the Land into a perversion of what it should be. If you are a fan of the books, the Sunbane will cause you great distress the first time you read about it. And ultimately, Lord Foul wants to literally destroy the world.
LOTR is richer in the sense of an encyclopedic history, functional languages, logical progressions from age to age and between races and types of beings. But it's poorer from an emotional standpoint.