Star Trek II showed up well before the age of digital filmmaking, so it was old-fashioned film for Nicholas Meyer and his crew. There were other limitations too, most of which had nothing do to with the fact that the most powerful computer at the time could probably be lapped a thousand times over by your cell phone.
The original Star Trek: The Motion Picture was a reasonable hit, but it was also super-expensive, and the final numbers didn't match the producers' sky-high expectations. So when they started the sequel, the order came down from on high: trim that budget and make do with less.
The funny thing is, the comparative cheapness didn't matter a whole lot. Meyer's effects team saved their ammo for the big stuff—the thundering showdowns between the Enterprise and the Reliant—and found ingenious ways to cut corners on the other stuff.
The scenes with the Enterprise leaving space dock were simply recycled from the first film, while key models like Space Station Regula One were simply Star Trek I models turned upside down. Much of the action takes place on a limited handful of sets, and in fact, Kirk and Khan never actually meet up face to face.
Most people never noticed. The Wrath of Khan scored its juice by returning to the strong characters and interpersonal drama that made the TV show so beloved, instead of trying to blow everyone's socks off with a lot of expensive visual effects. Everyone saw Kirk, Spock and the gang doing what we wanted them to do—bicker gamely while facing down a seemingly insurmountable threat—and the effects acted to augment the background rather than become the purpose of the exercise.
There's a lesson in there: a special effect is only as good as your story. A good script means people won't care how many shots of those Klingon warships they saw in earlier movies. A bad one means that not only will we care, but we'll spend the rest of eternity pointing out the stinky in that endless echo chamber that is the internet.
The masterminds behind Wrath of Khan chose wisely.