So, isn't "What is the difference between a musical piece and a song?" an important question in this case?
Yes, it's another way of asking the same basic question.
I attended a lecture last year by a college prof who argued that...
I know there are plenty of professors who would disagree with that one (and of course plenty who would agree). There's no denying that starting in the 1750's, most composers preferred
songlike melodies. Music critics of the time noted that melodies should be tuneful, memorable, active without being busy, and above all natural. In fact, this is one of the central defining traits of the Classical era. (Note the capital C, I'm talking 1750-1800 here specifically.)
Nonetheless, the word "song" continued to be used only for, well, songs. Even though composer sought to be songlike in their instrumental pieces, they never called those pieces "songs". The question I'm asking is about the terminology specifically as applied to individual pieces. The "song vs dance" distinction is useful, but it's far more generalized.
Etymology time: the Latin verb "to sing" is
cantare, and the Latin-derived root "cant" (transmitted through Italian) appears in a few terms worth knowing, if you care about music history. A "cantata", in its original definition, is any piece of music with singing. The corresponding term for an instrumental piece is "sonata", where "son" derives from the same root as "sound", and refers to playing a constructed instrument (to produce sound, but not by singing). The definition of "cantata" and "sonata" evolved over time, but the distinction between music with voices vs. purely instrumental always remained part of the difference. There's no sonata with voices, and no cantata without them.
So when musicians started to prefer "songlike" melodies for their instrumental pieces, they started using the Italian word that means songlike -
cantabile - in their musical directions. Here's the most famous example of all time: Beethoven's "Sonate Pathetique" second movement, headed "Adagio cantabile" (slow and songlike).
The title for the whole piece is "Pathetic Sonata" in French. "Pathetic" here refers to pathos, as the opening movement is exceptionally stormy for its time. "Sonata" because there's no voices: it's a piano composition.
All of that is just some background of course, since the key is how we apply the word "song" to modern music. I've freely used "song" to talk about instrumentals many times, like I'm sure we all have. But nonetheless we still acknowledge "instrumental" as a special category of song - one that is
missing an element usually present, thus the special name.
The Pink Floyd track "Several Species" is a case that, like Mosh mentioned, isn't really about "song" at all. It's an edge case, pushing and prodding at our definition of "music" itself - like wind chimes, 4'33", or Killswitch Engage's cover of "Holy Diver".