First, let's go way back. Richard Taruskin in his monumental Oxford History of Western Music, talks about the music of Western Europe and its descendants as being "literate" music because it is one of the very, very few musical traditions to have developed a form of musical notation. In fact, the only good musical notation ever invented. Before this happened, roughly a thousand years ago, everyone played "from memory" in the sense that they did not read from music. Like fiddlers, blues musicians and the Beatles, everything was in your head. The development of notation to extend human memory, was a huge advance because it meant you could write down musical ideas that would otherwise have been forgotten. You could also write music that would be too complex to be learned by rote. Imagine trying to learn and perform a Mahler symphony by rote-learning! Music notation also enables a composer to write a film score and have it recorded for the soundtrack in only one music union "service" which is 2.5 hours long. Yes, nearly all the orchestral music you have heard in the theater was recorded while being performed once and once only. This is only possible because of the clarity and efficiency of our music notation and the sight-reading skills of specialist musicians. If you can write it down, they can probably play it. At sight!
I strongly suspect that after millennia of musicians always playing and singing "by heart", watching them play and sing by reading from a score was quite a novelty with a real coolness factor. I have always thought it a bit magical to put a score before a musician who has never seen it before and listen to them make it come alive, at sight. The child Mozart, on tour of the courts of Europe before his tenth birthday would read whatever music people put before him, at sight. It was one of the marvels of his talent.
But after a few hundred years of this, I'm sure it lost its novelty and around the early part of the 19th century a new tradition came to be: the solo recital, played from memory. The two people who were probably most instrumental in creating this tradition were Clara Schumann, wife of Robert Schumann and a virtuoso pianist, and Franz Liszt, an even more virtuoso pianist. The two things they launched were the recital played from memory and the recital of great works, i.e. music not necessarily composed by them, but by great composers. The idea of playing music by people that were no longer alive was itself an innovation. Until this point, a virtuoso pianist would almost exclusively play music that they had written. This was true right up to the generations of Mozart, Haydn, Beethoven and Schubert. But just a couple of decades later, we have Franz Liszt touring Europe and playing sonatas and symphonies of Beethoven, the latter in his own transcription.
If you went to see Mozart playing one of his own piano concertos, he would have been playing from the music!! The reason being that he probably wrote it the day before and they probably only had one rehearsal. If you are going to go out and play a solo recital or concerto from memory that implies a couple of things: first, that you have had the weeks or months necessary to memorize lengthy pieces of music and that there is a "canon" of great works of music that are worthy and suitable of such attention. Both of these ideas were ones that came along in the 19th century. Before then, I doubt that anyone, even, say, François Couperin playing his own ordres for harpischord, would have played from memory.
In my own career as solo recitalist I played music from a lot of different eras, from the earliest pieces for lute and vihuela, written in the early 16th century, when no-one played from memory, to the newest pieces of avant-garde modernism, where some pieces were really almost impossible to memorize and others were intended specifically to be played from the score. In between, the bulk of the repertoire was quite suitable to be played from memory. So that's what I did: the early music I would play from the score as well as some of the newest music, and the rest I would play from memory. It seemed an appropriate policy.
It is interesting to look at the early music publications because you can see how they were organized to facilitate playing and singing at sight. In modern scores for multiple voices or instruments, all the parts are put together and synchronized vertically. This "score" is for the use of the conductor, so he or she can see all the parts together. Then each player gets his or her "part", i.e. the individual line only. But in the 16th century, when reading chamber music was an after dinner amusement, the parts were printed so that people could all sit around a table and each read their own part. Here is an example from the first printing of John Dowland's Lachrimae or Seaven Teares scored for an ensemble of five viols with lute. On this page you can see at the top the "quintus" viol part, in the middle the "bassus" and at the bottom, the "cantus". On another page are the remaining three parts. The layout is so that people sitting around the table can each see their part, right side up. There was no score in the modern sense.
Now, let's hear that. Here is the Lachrimae Antiquae, the first piece and the one shown above. The players are Jordi Savall and his ensemble: