A clear source for this distinction is found in a gemara which discusses the prohibition on music due to mourning over the destruction of the Mikdash.
The gemara concludes that from the verse "Don't drink wine with song" (Yishayahu 24:9) we can learn that instrumental music is forbidden; from the additional verse "Israel, don't rejoice unto delight like the nations" (Hoshea 9:1) we learn that vocal music is also forbidden. From the order we learn that there is a greater stricture for instrumental music. And according to Rashi there is also a practical distinction, in that vocal music is only forbidden in a wine house whereas instrumental music is forbidden in general. (Gittin 7a. Mourning over the Mikdash limits secular music of modest content. Immodest songs are always forbidden whereas songs of praise to G-d are permitted even nowadays except during sefira.)
A later source which reinforces this distinction is the statement of the Maharil that it is improper to make a wedding without musical instruments, for these are the main way we gladden the bride and groom (Maharil Eiruvei Chatzeirot).
Yet there is another source which seems to teach the opposite. The gemara in Arkhin (11a) discusses the music of the Leviim and others which accompanied the libations in the Mikdash, and concludes that the primary music is vocal music; the instruments are there only to accompany and adorn the singing. So in the Temple service it is the vocal music which is most important! What's the difference?
There is much evidence to support the following explanation: Vocal music expresses joy, whereas instrumental music induces joy. Vocal music is "inside-out"; instrumental music "outside-in". This suits their nature as well: singing comes from inside of us, whereas instrumental music is from the instrument which is external to us; it is an instrument for creating rejoicing.
For example, the two examples we find in the Torah of vocal singing are the Song of the Sea and the Song of the Well.
In both cases, miracle which caused the people to rejoice, and subsequently they sang. In fact, in each case the Torah explicitly tells us "Then Moshe and the children of Israel sang" (Shemot 15:1); "Then Israel sang" (Bamidbar 21:17).
By comparison, here are two prominent examples of instrumental music: When Shaul was in a terrified and unsettled state of mind, he sought "someone who knows how to play the harp" (I Shmuel 16:16). And when Elisha sought prophetic inspiration, he said "Bring me player"; and when the player played then G-d's spirit settled on him (II Melakhim 3:15). In each case the playing was not an expression of a state of mind but on the contrary a means to bring about an uplifted state of mind.
A seeming exception is the statement of Shlomo that in order to entertain himself he acquired "sharim vesharot", which literally translates as "men and women singers". Yet Targum and Rashi explain that this term actually refers to kinds of musical instruments (Kohelet 2:8).
Now we can understand the difference between the various halakhot. In the case of mourning, the main prohibition is inducing rejoicing not expressing it. Likewise, the purpose of music at a wedding is in order to induce joy in the bride and groom. So in these cases instrumental music is primary.
But in the Temple, the rejoicing stemmed from the beauty and holiness of the Temple service. The music was meant to express this joy, not to create it. "Which service is through joy and a glad heart? It is singing." (Arkhin 11a) In this case the singing is primary, and the instruments are only an accompaniment.
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