This poem came to my mind today after many years because a group I sing with is singing the American "The Colorado Trail," which opens, "Weep all you little rains/ Wail winds wail,/ All along, along, along/ The Colorado Trail." It is an echo of the poem we're discussing, which is itself, as others have said, like a song or even (to me) like a haiku, in that a lot is going on between the lines and phrases of the poem that is left unsaid: in the white space, so to speak.
The poem's meanings are reinforced musically, sounding (especially if you hear it in Middle English, which sounds (to this untutored ear) something like Broad Scots.) Listen, speaking it aloud, to the alliteration of the 'w's and 'n's and the assonance of the short 'e's in "Westrn wynde when wilt . . blow," followed by the "oh" sounds in "thou blow," which is connected by the "t/th" combination of "wilt thou". It works almost as well musically in modern English or even American! The second line "(that) the smalle rain down can rain," picks up in 'down' the "ow" in 'blow' and picks up the pronounced 'n's in "westryn wynde" in 'rain.' Yet the vowels in that line create a flatter (wetter?!) sound. As others have said, the word 'that' is almost surely understood in the context; it acts something like the 'and' often used in Scottish Gaelic to express cause. The couplet almost explodes with frustration: When on earth will this dry season (or perhaps the stormy season of Northeast winds) be over, and the Western wind blow again, so that the small (soft, gentle, as opposed to fierce and stormy) rain can come?" (That lullaby, "Sweet and Low" comes to mind: "Blow him again to me . . ." sung by the wife waiting with her baby for her man to sail home.)
There is a space between the two couplets of the poem in which the frustration takes a turn. "CHRIST! That my love were in my arms/ And I in my bed again!" Again, hear the assonance: Christ/my/ arms/ I/ my/ again. The connection between the two couplets is the frustration this (surely) man is expressing so eloquently. At a time when the sea connected people and places, when the waters were the highways far more than the roads on shore, especially in Britain, the winds were often what made the difference between being able to be home or being stuck in some other place or at sea, and the longing this poem expresses is made vivid to us by the music of the words in the rhythm and rhyme that is, truly, like a song. I actually do not know the Peter Pears (20th century) song, and I'm sure it does the poem credit, but I cannot see that it could improve upon it. Now, having said that, the melody of "The Colorado Trail" is very beautiful and in my opinion adds to the haunting refrain of what is a lament for a lost love. I wonder whether the person who wrote that song knew this poem!