Fundamental problem: social networks with no-one on them are not fun and not sticky.
Fundamental problem 2: if anyone can read what you're writing, it's not a social network, it's a blog. So you can't just let anyone see what people are writing.
The way FB handled that in the early days was that by default, people in the same university network could see your status etc. I don't remember exactly what they could and couldn't do but the point is that they created a semi-privileged circle of people with some access based on a pre-made group that people would be familiar with - fellow students at the same university.
That gave people a network from day 1 in what felt like a controlled way. Also, as noted, they started with a desirable closed group that people wanted to belong to.
They would later expand to other Ivy league schools — aka networks — one by one, rather than simultaneously, as a way to slow their growth.
I imagine an unintended side effect of this growth strategy was the sense of FOMO that it engendered among university students that had heard about but couldn’t join the exclusive/elusive Facebook, unlike the other social networks like Friendster, Hi5 etc.
For each network on their expansion list, Zuck & team would furiously scrape the student directory to seed the new network with plausible student info.
It wasn't just that. At the beginning, you had to be at an Ivy League. And that was the problem, G+ was invite only among a bunch of nerds, whereas FB was invite only among people who network for a living.
If anything, the real distinction is that a network for college students is appealing to teenagers, who are the perfect audience for social media. A network for adults doesn't draw in this crowd.
Unfortunately this increased the barrier to entry at launch too, which for social networks whose value is tied to users generating media is a cardinal sin.