Q:

Why aren't the Valar considered gods?

As an avid Tolkien fan and an increasingly active member of this site, I cannot count the number of times I've seen people refer to the Valar or Maiar as gods, only to be corrected by other members (often of high reputation) that they are more like angels, etc.
Why?
Even if the Maiar are more like angels (a Gabriel/Gandalf comparison has a lot of merit), the Valar are almost exactly gods, taken from Greek or Norse mythology with barely even any alteration. They have specialties and spheres of influence, they are supernatural beings of incredible power, they are immortal (at least to the extent that Greek deities are), they live in a magical land just like Asgard or Mt Olympus, they were involved with the creation and shaping of the world before retreating back after the coming of men, etc. Aule in Valinor is practically interchangeable with Hephaestus on Mt Olympus, Yavanna with Demeter, Ulmo with Poseidon, and on and on and on.
Obviously Eru Ilúvatar rules over them as the utmost power in Arda, but hierarchical relationships are well-known in polytheistic traditions. Tolkien himself was a Christian, and he famously kept Middle-earth from having too much religious flavor in-universe, but he must have known that a lot of his creations would have been heretical if they'd been presented as more than fiction (e.g. "God gave dominion over the Earth to men, not men and elves!" or "God required no input from his angels to create the world!"), so why would this particular distinction be so important?
Where does this tradition of the Valar being absolutely, positively not considered gods come from? Did Tolkien himself insist on it? If so, did he ever comment on why he patterned them so clearly off of a polytheistic tradition (i.e. specialized nature-deities with their own kingdom, desires, and familial relationships), rather than a more monotheistic structure (i.e. angelic spirits of goodness without independent lives, serving as messengers of a single authority)?

A:

They are.
That's a bit of a simplistic and sweeping statement, so I obviously need to expand some more.
First and foremost, in all of the early Silmarillion work, up to and including the 1937 Silmarillion (which Tolkien had sent for publication before writing LotR) the word "gods" is used for the Valar very liberally.  I'll add some quotes to illustrate (the capitalization of "Gods" here is editorial and should not be seen as significant):

Of all  things which  the Gods made  they  have  most  renown, and  about their  fate all  the tales of the  Eldar are  woven.
she set the  crown of  seven mighty  stars to swing,  the  emblem  of  the  Gods,  and  the  sign  of   doom.
Thus  it  came  to  pass  that  after   long  council   the  Gods resolved  to  make  an  assault upon  the fortress  of Morgoth  in the North.
In  the  North   these  shores,   in  the ancient  days  after  the  Battle  of  the  Gods,  sloped  ever  westward...
Yet  Morgoth  whispered  that  the   Gods  kept   the  Eldar captive,  so  that  Men coming  should defraud  them of  the kingdoms of Middle-earth...

In the 1950/1951 revisions of the Silmarillion, carried out before publication of LotR, Tolkien began the process of removing references to the "gods" from certain works, but in a rather piecemeal fashion, and they still remain in the final versions of many of the stories; for example, the following quote from the last chapter:

...the host of the Gods were arrayed in forms of Valinor...

Compare with the published version:

...the host of the Valar were arrayed in forms young and fair and terrible...

So the conclusion is that the removal of references to "the Gods" was not an authorial one (i.e one made by JRRT) but rather an editorial one (i.e one made by CT).
Even so, references to "the Gods" do remain in the published Silmarillion, and I'll quote all of them that apply to the Valar:

The Great among these spirits the Elves name the Valar, the Powers of Arda, and Men have often called them gods.
...westward shimmered the fields and pastures of Yavanna, gold beneath the tall wheat of the gods.
Thither we cannot come where the Gods dwell in bliss.
Which of you has seen the least of the Gods? Who has beheld the Dark King in the North?

So in summary, there is no "tradition of the Valar being absolutely, positively not considered gods".  While the use of the word "Gods" to refer to them did drop out in stages, it was never fully abandoned, and even despite CT removing most of the references from the published work, he still retained a few.
Tolkien's actual intentions in this are given in various places throughout his Letters, so I'll quote from Letter 200 to provide one example:

...when the Creator made it real ... they desired to enter into it, from the beginning of its 'realization'.
They were allowed to do so, and the great among them became the equivalent of the 'gods' of traditional mythologies; but a condition was that they would remain 'in it' until the Story was finished.

So your observation that "the Valar are almost exactly gods, taken from Greek or Norse mythology with barely even any alteration" is in fact correct: that was Tolkien's intention, but with a modification to the theological framework that - while they were certainly involved in the creative work - they were agents of it rather than creators themselves.  Any tradition of the Valar not being "Gods" comes from later interpretations by third parties, but has neither authorial nor editorial basis.

A:

Because the Valar resemble angels (servitors of a higher power) more than they do independent gods with unique desires (and the power to create what they like) as most pantheistic religions do. This does not mean they did not have independent thought or will, but their will was always second to Ilúvatar's.

There was Eru, the One, who in Arda is called Ilúvatar; and he made first the Ainur, the Holy Ones, that were the offspring of his thought, and they were with him before aught else was made. And he spoke to them, propounding to them themes of music; and they sang before him, and he was glad.
But for a long while they sang only each alone, or but few together, while the rest hearkened; for each comprehended only that part of me mind of Ilúvatar from which he came, and in the understanding of their brethren they grew but slowly. Yet ever as they listened they came to deeper understanding, and increased in unison and harmony.
And it came to pass that Ilúvatar called together all the Ainur and declared to them a mighty theme, unfolding to them things greater and more wonderful than he had yet revealed; and the glory of its beginning and the splendour of its end amazed the Ainur, so that they bowed before Ilúvatar and were silent.
Then Ilúvatar said to them: 'Of the theme that I have declared to you, I will now that ye make in harmony together a Great Music. And since I have kindled you with the Flame Imperishable, ye shall show forth your powers in adorning this theme, each with his own thoughts and devices, if he will. But I win sit and hearken, and be glad that through you great beauty has been wakened into song.'
REF: From the AINULINDALË, first chapter to The Silmarillion

The simplest answer might be best: Because the Ainur lacked the capacity most gods have in almost any other pantheon. They could not bring things to life without Ilúvatar's blessing of Life.

The Dwarves were created in ancient times by Aulë, one of the Valar. Aulë was a smith and a craftsman and he wanted to teach his skills to the Dwarves. He made Durin first and then six others. But Eru had intended the Elves to be the Firstborn race so he commanded Aulë to put the Dwarves to sleep until after the Elves awoke. Eru gave each of the Dwarves their own souls so they could exist as independent beings. At first, when Eru had scolded Aulë for his creating the Dwarf Fathers, Aulë was going to destroy them, but Eru stopped him, noting that they now had their own souls. The Dwarves are thereby the "adopted Children" of Eru.

This lack, more than anything else, caused the rift between Melkor and the rest of the Ainur. He wanted to create Life on his own without needing sanction from Ilúvatar.

They were partners in the Music, but their contribution was directed and supported by Ilúvatar's will.

A:

I like Thaddeus' answer as an in-universe answer, but I found some evidence to support an out-of-universe answer.
It seems as though Tolkien himself never conceived of the Valar as gods, but some Elves did hold that belief in early stages of the story. The Book of Lost Tales, volume on of HoME, has a short text between two Elves called "Link between Cottage of Lost Play and Music of Ainur". It starts like this:

'But', said Eriol, 'still are there many things that remain dark to me. Indeed I would fain know who be these Valar; be they Gods?'
'So be they,' said Lindo, 'though concerning them Men tell many strange and garbled tales that are far from the truth, and many strange names they call them that you will not hear here.'

However, later on in the same text:

'Yet,' said Eriol, 'tell me, Rumil1, I beg, some of what you know even of the first beginnings, that I may begin to understand those things that are told me in this isle2.'
But Rumil said: Iluvatar was the first beginning, and beyond that no wisdom of the Valar or of Eldar or Men can go.'
'Who was Iluvatar?' said Eriol. 'Was he of the Gods?'
'Nay,' said Rumil, 'that he was not, for he made them. Iluvatar is the Lord for Always who dwells beyond the world; who made it and is not of it or in it, but loves it.'

The published editions of other works, including The Silmarillion, refer to the Valar as "the Powers". I'm not going to dig up a quote for that because there are loads, but it's clear that Tokien decided to change his terminology, even if he never thought of the Valar as "gods" in the same way Iluvatar was a god.

1 Rumil is presented as a very ancient Noldoli, at this early stage translated as "Gnome". Christopher Tolkien's notes indicate that the Noldoli eventually became the Noldor. It's likely that Rumil learned what he knows from the Valar directly
2 I believe the isle in question is Tol Eressea, but I'm not certain. I'm sourcing a crap ebook, and it's not easy to read.

