As previous replies have pointed out, advances in hardware were key. In 1962, integrated circuits were still in their infancy. They had only been invented four years earlier, and the only ones in production were being built for U.S. military projects like the Minuteman nuclear ballistic missile. And even those were very small-scale circuits, with only a few logic gates per chip.
Computers like the
PDP-1 were built using thousands of discrete transistor components for their logic and magnetic cores for their main memory. The price for a basic PDP-1 at that time was around $100,000 in 1962 dollars, equivalent to about $800,000 today. That's a *basic* system; the point-plotting CRT display used in
Spacewar! would have added quite a bit to the cost. The machine with all its peripherals took a good fraction of a room and probably weighed at least 2000 pounds. And running Spacewar! pretty much consumed the PDP-1's entire processing power. (Since the display was point-plotting only, the spaceships had to be drawn as series of dots, and the display had no storage ability, so a lot of processing overhead was needed to constantly refresh the entire list of currently displayed dots.)
When Spacewar! was written, the video game was basically a science-fiction concept, and computer graphics itself was just beginning to develop. Arcade games at that time were purely electromechanical games, such as pinball. The first commercial arcade video games (
Galaxy Game and
Computer Space, both of which were ports of Spacewar!) didn't appear until 1971;
Atari's Pong came out the following year. Arcade video games of the early 1970s used custom state machines built from TTL logic chips instead of programmed computer systems; the first microprocessor-based arcade video games appeared starting in 1975 with
Taito's Gun Fight, which used the Intel 8080. It was those programmable microprocessor-based systems that really allowed video game development to take off; for example, Asteroids was based on a 6502. Incidentally, Asteroids' vector display system first appeared in an arcade game with Cinematronics'
Space Wars in 1977.
Spacewar! was widely ported to various computer systems during the 1960s and 1970s, so it's no surprise that Asteroids bears a strong resemblance to it.