Note: It sounds like the functionality of the radio room has changed so that this answer no longer really applies. I won't be updating it since I don't play the game anymore.
A few people have pointed out some advantages I hadn't thought about, which I found interesting since I've found the radio station to be not only worthless, but harmful.
- Pregnant women can't deal with incidents, so more of your workforce is available with a radio room. This is easy enough to manage by not having too many women pregnant at a time, and not filling a single room entirely with pregnant women.
- Dwellers arrive at the vault ready to go, unlike children who must grow. Again, this is easily manageable by not having too many women pregnant at once.
- Increases happiness. I haven't found this to be true in practice. If it is true, the effect is negligible. It's important to note that procreation raises dwellers' happiness to 100%, so pregnancy raises happiness as well. I've actually found it to be much more effective at increasing happiness than the radio room.
- The radio room can bring in a dweller in a couple hours, whereas a child takes 6 hours. This is true, but you can produce more people faster and more reliably through pregnancy by having just a couple women pregnant at a time. You can also attempt to engage in eugenics and make extra awesome people.
So there are a number of minor advantages which are easily dismissed if you manage your pregnancies correctly. On the other hand, there's a large drawback: you can't turn it off. If you decide you have enough people and don't have a radio station, you can simply avoid putting anyone in the barracks. But, if you do have a radio station and don't want anymore people, you have limited options, and they all kind of suck:
- Destroy the station. Build another one when you need it again.
- Destroy additional living quarters. This requires you to have exactly the correct amount of living quarters for your population all the time. You can't randomly upgrade one for a challenge, or anything like that.
- Let a line of loiterers build up at your door. This is super annoying.
- Let some dwellers die (and remove them). Maybe that's your thing. Of course, the radio station will attract replacements, so you'll have to kill them too.
- Leave the radio unstaffed. The room still draws power, and can still have incidents in it, which is a bigger hassle when there's no one in the room to deal with it.
Taking this into account, I find an empty hole in the ground to be more advantageous than a radio station.