The primary advantage of an integrated circuit is not that the transistors (switches) are miniscule (that's the secondary advantage), but rather that millions of transistors can be created and interconnected in a mass-production process. All the elements on the integrated circuit are fabricated simultaneously via a small number (maybe 12) of optical masks that define the geometry of each layer. This speeds up the process of fabricating the computer -- and hence reduces its cost -- just as Gutenberg's printing press sped up the fabrication of books and thereby made them affordable to all.
The IBM Stretch computer of 1959 needed its 33 foot length to hold the 150,000 transistors it contained. These transistors were tremendously smaller than the vacuum tubes they replaced, but they were still individual elements requiring individual assembly. By the early 1980s this many transistors could be simultaneously fabricated on an integrated circuit. Today's Pentium 4 microprocessor contains 42,000,000 transistors in this same thumbnail sized piece of silicon.
It's humorous to remember that in between the Stretch machine (which would be called a mainframe today) and the Apple I (a desktop computer) there was an entire industry segment referred to as mini-computers such as the following PDP-12 computer of 1969:
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