I have to go with brain surgeon. It’s technically difficult, it’s physically difficult, it’s emotionally difficult, and it requires way more people skills than “rocket scientist” which probably requires almost none.
As for me, I said brain surgeon. Mainly for the reasons mentioned above: Medical school is very competitive, as aerospace engineering isn’t quite as selective. Also, neurosurgeons have a human life in their hands; there are no do-overs.
To be fair, rocket scientists do at times also have human lives depending on them doing their job properly. It’s just not as immediate and is more emotionally distant.
But space travel is completely optional. No one is going to die because they can’t catch the next rocketship.
Ah, but it’s not voluntary if a NASA launch were to go off course and crash in some Florida neighborhood. 
It’s not just aerospace where the lives of people depend on the work of rocket scientists. Rocket scientists research and design RPGs, MLRSs and ICBMs.
The intellectual answer is “brain surgeon.”
But brains and surgery are icky and gross, and rocket science is way cool. So the emotional answer is “rocket scientist.” 
But ask me again when I have a brain tumor. :eek:
Someone important to you may die if one of the many rockets used by the US military don’t work properly when needed. And life would be a lot shittier without the constellation of man-made satellites that provide us with GPS, satellite radio, earth imagery/telemetry, and so on. All of that depends on well-designed rockets performing flawlessly.
I think a lot of what a brain surgeon does is a matter of repairing or modifying some basic stuff, and then getting the hell out of the way so that the brain can finish fixing itself. The success or failure of brain surgery is only partially owned by the surgeon.
In stark contrast, a malfunctioning rocket rarely fixes itself. And if it does, it’s almost certainly because the rocket scientist designed it do so. The success or failure of a rocket belongs almost exclusively to the rocket scientist.
I therefore conclude that a successful rocket scientist is more impressive than a successful brain surgeon.
But rockets, as complex as they are, are much less complex than the human brain. You say that the surgeon is fixing some basic stuff, but I think that’s incorrect. It isn’t like the brain is one homogeneous mass of neurons that can all take over for each other. As an example, if you’re removing a tumor near an optic nerve and are off by just a very tiny bit, you can destroy or degrade vision in one or both eyes.
Back in the day, rocket scientists were going to the Moon or threatening mass destruction of the planet, doing all their calculations by sliding around a couple of sticks. Compare that to brain surgeons, who were lobotomizing people or otherwise just drilling holes and poking around to see what happened. I think you’d have to historically give the edge to rocket scientists.
Now, though, the rocket scientists have their fancy “computers” to do all the hard calculating, and brain surgeons are, I do believe, actually helping their patients. So I think it’s tipped the other way, towards brain surgeons now.
No rocket scientist I remember was ever featured in Reader’s Digest. I remember two brain surgeons, Epstein and Cerullo. Epstein (Aaron Alligator) got one feature and a book section.
Sometimes the brain surgeon and the rocket scientist both lose to the grammarian.
Forgot to mention: surgery is now done by micrometer accurate robots, designed by rocket scientists… Soooo…
I was divided on this, but ultimately went with brain surgeon. The final deciding factor was the addition of physical coordination necessary for the surgeon. Otherwise they are equal to me.
Brain surgeon, because it’s a more useful skill. Saving someone’s life is the most impressive thing there is.