So, now to answer what real theorists and experimentalists are like:
Just to give you some background on where this answer comes from: I am a Ph.D. student in physics, about to finish my Masters in Physics. I spend all day, every day surrounded by physicists, experiment and theory, so I think I can at least give an overview of what each discipline is like. Personally, I am an experimentalist.
Any real physicists, in either discipline, needs to have an understanding of the other. The Theorists in the high energy group in my department spend a lot of their time talking about what we will see in the LHC. In order to do this they need to know how particles are detected at the LHC. They don't need to be able to put together the ATLAS detector from memory, but they have to be familiar with the experimental methods, otherwise, they can't really give any productive insight into what new discoveries will look like.
On the opposite side of the coin, a good experimentalist must have an understanding of theory. A high energy experimentalist has to be well versed in Quantum Field Theory. If they don't understand how elementary particles will scatter, how will they ever be able to design good detectors? Thus, to imply that experimentalists are essentially monkeys with wrenches "counting electrons on a screen" is just plain wrong, not to mention insulting.
Now what skills will one develop when they become a theorist:
Obviously, being good at math is important, but there is much more to real theoretical work. You will most likely not be working with the fundamental equations of the universe! You'll have to learn approximation methods, which is arguably not the most interesting thing one learns in theory! You'll have to be able to use perturbation methods is all fields. In condensed matter theory, you'll spend time learning approximation methods like Hartree-Fock, density functional theory, etc. etc.
You will have to learn numerical and computational methods to help you produce results. Few non practicing theorists realize how important computers are to modern science. Whether you are doing simulations of protein folding or Lattice QCD, computers are essential to modern day theoretical work. Expect to spend a lot of time learning computational methods as a theorist.
It's not all cups of tea and blackboards!
What skills do experimentalists need?
Experiments that involve a person physically counting dots on a screen do not exist is modern day physics. Anyone who says that that is what goes on in a real physics lab might as well be saying that experimentalists spend their days rolling carts down ramps and measuring g over and over again!
Many experiments involve lots of automation. Expect to learn how to interface devices (oscilloscopes, lock in amplifiers, motion controllers, laser oscillators) with computers. You will also get very good at programming in experiment!
Chances are you will need to build many of the components for your experiment yourself! No one has done that experiment before, why should anyone sell the stuff you need? It's quite possible that you'll need to take a machine shop class along with that quantum field theory course next semester!
You've worked through all the theory from QED and understand how stimulated and spontaneous emission arise from the quantization of the electromagnetic field? (You will do this in a graduate quantum course, regardless of discipline.) That's great, but can you build a laser cavity and tweak the mirrors such that you get your oscillator to mode lock so you can get the femtosecond pulses you need? You will need to be able to understand how real lasers work. Trust me, it goes way beyond stimulated and spontaneous emission!
Also, I won't lie, expect to do some grunt work! Can you replace piping or a water pump when a cooling system breaks down? Effectively transfer liquid helium? How about soldering? Designing circuits?
Experimentalists learn WAY more than just Quantum Mechanics and E&M in grad school! Some of the experimentalists I have met are they most mathematically able people I know, and they are also some of the most well rounded, capable people I've ever met!
The moral is that both experiment and theory require much more than physics lectures imply! It's hard work, but very rewarding if you enjoy it!
Finally it can't hurt to post another link:
https://www.physicsforums.com/showthread.php?t=240792