By default you have 9 happy faces to work with, plus 1 for each natural wonder you find, plus 4 for each luxury good you have. Then you can also get some extra happy faces through policies or through local city happiness buildings. Every city you found knocks 3 happiness off of your bottom line. So in a sense every city you found kills off the potential for 3 people in your capital. People in your capital tend to be better than people in other cities and right now we're replacing them with just the base tile of a new city so this is a pretty terrible trade. If the new city gets you a luxury that's actually a net positive though since you get the city and the first population completely for free.
Another difference comes from the whole local happiness thing. If you're able to build a colisseum, a circus, a pagoda, and a mosque in the new city then you've got the local happiness to pay for 7 people in that city that you couldn't otherwise pick up. So you subtracted out 3 people from your capital and got 7 people in return. Each individual person may be worse but you've more than doubled how many of them you can have. You do tack on a 5% science penalty and a 10% culture penalty for each new city but with just a library, monument, and the two religious buildings you will be bringing in 7 culture and 10.5 science from the city. It feels like that has to more than cover the extra cost unless your capital is really insane.
Now, I can definitely envision ways to have your capital be really insane. If you go completely down the tradition policy tree your capital's people will only cost half an unhappiness each instead of the full unhappiness in a regular city. This means you'd need to trade 6 people in your capital for 7 in another city. Your capital is also likely to build a bunch of multipliers unavailable to the rest of your cities (like 50% more science from national college) which can really make the 6 people in your capital worth an awful lot. I can see settling more cities to pick up luxuries since those are happy face positive and the extra safety you get from being able to produce more military units is likely worth the policy and tech hikes. Especially since unless your capital is making 200 science or 70 culture you're eventually going to be outpacing those cost multipliers.
In my current
For example, I think a policy with a single city would cost me 413 culture. I'm making 16 culture in my (terrible) capital so that's 26 turns for a policy. Each city I found adds an extra 41 onto the cost of the policy. So each city has to make at least 1.6 culture per turn to offset the 10% culture cost. Considering they're guaranteed to make 1 culture per turn because I have the liberty opener... I should have as many cities as I can get my hands on! Build a monument and go! Especially with a religion like I have with the ability to buy 2 buildings for 3 happy faces in each city.
I don't think I can get any wonders because I don't have any single city that's any good but I have lots of cities that are decent. I expect I should be able to hold off any invaders by just setting all of my cities to build units. But I am last in hammers, so maybe that's not even true. I really need to get some policies in the exploration tree. Unlocking 3 hammers per coastal city and 3 happy faces per coastal city (that has built a light house, harbour, and seaport anyway) looks to be really, really good. Then I can get most of my cities up to size 10 for no extra unhappiness! Snag a zoo and I can get up to 12! Of course most of these cities don't have anything resembling 12 tiles worth using. Maybe I'll eventually build buildings for specialists so I can do better than 'unemployed dude for 1 hammer' which I've had to use a few times already this game. (That population growth was a mistake!)
I'm not sure about saving up for a golden age. It would be pretty sweet but it takes so many happy faces and I'm really not that far above zero. I could stunt all my cities growth instead of just most of them to try to force it to happen. 20% more production and culture would be pretty sweet. Maybe that's how I get the policies together to work on exploration... Hmm...