Just circling back on this as I've been doing some academic research on this... out of a list of 67 authors from the UK and North America writing predominantly in the 20th century, 21 listed Tolkien as an influence and 15 R.E. Howard.
Which feels closer than one might expect.
The answer to that to a certain extent is that personal influence isn't the only influence. Epic fantasy authors like Erikson and Esslemont, Feist, Eddings, might poo poo the notion that Tolkien influenced them, but he influenced the art form they chose to use. Erikson might list Howard as an influence over Tolkien, but when he is also listing Donaldson, Le Guin and Robin Hobb, he's receiving Tolkien influence through all of them.
But of course the more you play that game, the more Howard grows too. There's 14 authors on the list counting Fritz Leiber as an influence. Sure, some of them such as Erikson, Ian Esslemont, Bill King count Howard as an influence too (okay, that's all of them). But you get a group of authors like Patricia McKillip, Tim Powers, Tanith Lee, Feist and Tad Williams (and plenty more) who now show a Howard influence, some of them that you mightn't expect it from.
I am beginning to suspect the answer to Tolkien's overbearing influence though lies primarily with fans than with the actual creatives. Yes, he is hugely influential there, but not overbearingly so. We are the people who go "ahh! it looks like Tolkien". But as we should all know, when you go looking for something, you find it.
Anyway, I'd like to do more research on this before declaring anything definitive, but in terms of actual authorly influence rather than dominating cultural and commercial mindsets, I suspect it's actually a fight worth talking about, insofar as this is a fight. Cultural and commercial remains decisively with Tolkien of course; in terms of shaping terrain, Howard is like a storm that caused some trees to crash over a while back, and Tolkien is the trees you can still actually touch. Both great, but the one you can see commands primacy.
I also suspect that by now, finding a fantasy author - certainly in the trad field, but really just any field - who doesn't have influences that are at some point indebted to both would be more unusual than finding one who has both.
Also in terms of what Howard brought to the table, I think everytime you see someone marry horror influences to action, or to the historic you see a bit of what Howard wrought - the marriage of pulp action to Lovecraftian (and Poe-esque and Dunsanian) horror and Burroughs/Haggard lost worlds, with a lot of historical novel additions. Which is perhaps why Howard seems to be a bigger influence on the start of Urban Fantasy than Tolkien.
I think also it's unsurprising Howard appeals to more cynical authors. I saw someone refer to Howard as believing in the world as never changing; he one hundred per cent didn't. He lived in Texas at the time of an oil boom that radically and violently reshaped his home, while having to constantly move due to concealing the nature of his mother's illness. He lived a transient life marked by much melancholy. Conan is in part his revenge story, his tale of how the barbaric values of the frontier were greater than the civilised values of the oil boom, but also an elegy for a dying world. Middle Earth gets restored; Conan's world is to be changed, even if it not in his lifetime, and most of his great deeds bring no lasting good to himself. He is a natural influence for any author equally pessimistic about the coming world, or skeptical of the good of civilisation. Not unlike Tolkien there.
