Increase population with Radio Station / Advantage over pregnancy?

In addition to what's already been mentioned, it is possible to be both pregnant and working in the radio studio. Other comparisons:

• Pregnant women cannot defend against fire, radroaches, or raiders, so if you have 50% of your vault pregnant, you are very vulnerable.
• Children consume resources while they are growing up without contributing to the vault. Having fourteen kids around will drain you dry. Dwellers attracted by the radio are ready to work as soon as they arrive.
• In true Bethesda style, I don't think children can be killed, which means that a bad raider attack could leave you with a vault of just kids, fun.

The Radio Station gives you a full grown Dweller ready to work with only a few hours waiting between Dwellers, if the RNG is in your favor. With pregnancy, you have to wait for the female Dweller to get pregnant, deliver the baby and then the baby has to grow. This takes 6 hours (3 for pregnancy, 3 to grow in adults) compared the Radio Station (only 2 hours) but the former is 100% reliable while the latter may or may not work.

Edit: Adding more Dwellers to the Radio Station will make it operate faster, while upgrading the Radio Station and merging it will increase the chances of attracting a Dweller

The other answers are good here, but so far no one has mentioned another benefit of the radio station: It increases the happiness rating of all the dwellers in the vault.

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.

Advantages:

• 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.

I've had the radio station for two and a half days and I've received 3 dwellers with it so far. I only have 1 radio station that's been upgraded and it has 2 dwellers in it. I must be really lucky to have been so successful with it so far but I think it's great. I still make some of my woman pregnant but I think it's really important to try and get new dwellers via wasteland cause it's never bad to have a new blood line in your vault. Especially since a daughter can't mate with her father and a son can't mate with his mother so I almost ran out of available men and woman before the radio station.