Excellent points about the distinction between the "legendarium"--the
assemblage of epic tales Tolkien spun over the course of a lifetime--and the
novelistic LOTR.
The "mythic" elements of Tolkien reside more in the Sil and in the many
fragments, revisions, and rewritings of the whole HoME series. Many first
time Sil readers complain that it's like the Bible which lacks concrete detail
and narrative cohesion which provides it with what for lack of a better term we
can call "human interest" (or hobbit interest; Tolkien was deeply worried that
his loyal readers would balk at the lack of hobbits in it if it were
published). To all that I say yes, the Sil is vague because that is what
myths are like. All mythological writing, the details are sketchy, there
are revisons, contraditions, embellishments, etc., to the basic tales.
Myths are passed down from multiple sources, and always vary; look at how
Classical Greek myths come down to us from Homer through other Athenian writers
and then from Roman sources. Tolkien achieves much the same effect,
perhaps simply through his tendency to nit-pick his stories and his notorious
inability to declare a story "finished." The rough, ever-changing
contours of his broad mythic roadmap--the music of Iluvatar as it unfolds
throught the earliest ages of Arda to the final end (too pressed for time to
link to NZ Strider's repost of the eschatology thread)--form the basis of the
"mythology for England" that Tolkien was creating. Now I must add that
this mythology is really a wholly artificial one in that it is the
product of one man's eccentric writing habits and not the natural accumulation
of centuries of oral and then written tradition and narratives. Yet the
fact that a single Oxford don could replicate something like the mythic output
of ancient civilizations in his spare time between tutoring students is what
always amazes me.
Now as Curious very rightly points out, LOTR is a hybrid of the kind of mythic,
epic narrative found in his other fragmentary writing with a 19th-century
novel, so it is not really "mythic" in itself. It is the sudden, violent
intrusion of myth into the workaday world of realism (best embodied by those
little surrogate Victorian Englishmen, the hobbits). The mythic
superstructure of Middle-earth is only dimly discernible throughout LOTR.
LOTR is also laden with linguistic and historical detail to give it the
illusion of a living world, and in that sense differs from myth as well.
Think of it like the cuneiform tablets of ancient Sumer naming their rulers, or
the accounts of battles preserved in archives or monumental inscriptions,
versus the Sumerian epic Gilgamesh. LOTR "fleshes out" the "modern"
history of the Arda of the lengedarium, leaving the earlier ages to the realm
of myth (note how much detail we get in the appendices about the Second and
especially the Third ages versus how little we get of first Age).
I will leave the discussion of Beowulf and Celtic mythic to others who know
more than I do and are qualified to speak on that topic. Tolkien's view,
though, was that England's native mythology had been lost and much of what was
preserved, like the Danish Beowulf or the French Arthurian legends, were not
indigenous to the land. It does take a lot of hootzpah to think that you
can single-handedly weave a mythology for your land to make up for millennia of
lost legends! :) That is either inspiriation, or
megalomania!

"'Never laugh at live dragons,
Bilbo you fool!' he said to himself,
and it became a favourite saying of
his later, and passed into a proverb."
--The Hobbit