I was talking to my piano teacher last night about this and we agreed that the term 'classical' is a layman's term and actual 'classical' music is really only from the late 1700s to the early 1800s (preceded by baroque and succeeded by romantic). In which case is there an agreed upon historical term for this period? If the Renaissance is considered to be 1300-1700, is there a name for the period of 1685 (Bach's date of birth) to 1975 (Shostakovich's death)?
Also is there any agreed upon explanation for why this massive increase in compositional sophistication happened? In conversation with my teacher, I suggested perhaps it was thanks to the invention/perfection of keyboard instruments such as the harpsichord, pianoforte, piano and organ.
Let's piece down your questions bit by bit. Classical music is generally used as a name for the entire genre of music that is generally orchestral or featuring orchestral instruments. It's not just a layman's term, and is used frequently by scholars as well.
As for eras, you have your dates a bit wrong. In music, the Renaissance is generally ~1400-1600, then you have the Baroque era from 1600-1750, then Classical from 1750-1830, Romantic from 1830-1900, then 20th Century.
As for why compositional sophistication began, there are a lot of factors. For a lot of the 20th century in Western Europe, most music was being made by the church, because they had the most money and means to hire composers and musicians on a regular basis. However the church didn't allow instruments into churches or religious services until the pipe organ in the ~1500s (instruments were seen as to earthy and lustful, whereas the human voice was seen as a form of prayer, even in song). So, in the ~1500s you see this huge upsurge in organ music. And the organ, being a huge and complicated instrument that can be played with 2 hands and 2 feet, capable of all sorts of sounds, you see the compositions starting to reflect that. At the same time, you see lots of innovation in instruments. The Baroque era saw some huge changes to instrumentation, and developments of new instruments. Renaissance instruments, like the flute, which had remained unchanged get total overhauls, and become capable of playing in different keys, playing accidentals, tuning to other instruments (something that is impossible when you're playing just a single piece of hollowed wood), and more. You see a huge uptick in the use of brass instruments, and keyboard instruments start being developed, the most popular of which was the harpsichord. You also see the start of opera and other secular song forms. Along with that is he development of singing schools, and the first real opera training academies with the advent of the Castrati.
So why did this all happen? I don't think there's any historical consensus, but there are many factors - the aforementioned church opening its doors to instruments, the interest in wealthy nobles to become arts patrons en masse, the development of technology and raw materials to allow the contruction of instruments like a pipe organ and others, the increase of composers and musicians being hired, many of whom were also instrument makers, the desire amongst the nobility for constant novelty and therefore the pressure to continue innovating, the post-Rennaisance nobility wanting more culture and social events with musicians, the nobility paying for lavish houses and parties instead of paying for a military regiment... I doubt you could point to one factor, but social culture, technological innovation, and the trend of beginning urbanzation would all likely have contributed.