Which is best: studio portraits or natural outdoor portrait photography?

Neither. It all depends on what mood you are after in your photos. I use studio for specific types of photography. Inside the studio I shoot fine art nudes, calligraphy body art, and dramatic portraits or portraits I use for photoshop manipulations. Outdoors I shoot urban portraits, natural light portraits, where I have the advantage of choosing from variety of backgrounds. On the other hand, I can create dramatic light outdoors as well with the help of flash photography. By combining natural light and flashes I am able to capture images that would not be possible to create in ambient light.

I find myself shooting outside more often, because I like the endless possibilities of what surrounding world has to offer. Whether it is city or nature the options are limitless. Studio photography usually has a very stiff and artificial feel to it, but it doesn't have to be so. It all depends on how skilled are you with controlling the light and how good of a connection you have with your model. Outdoors and indoors portrait photography come with challenges that you will need to face. Outside you have to be in control of the background, and also very skilled with understanding the ambient light and how to bend it to your will. It is much harder to control ambient light than artificial studio flashes.

When you choose between indoors or outdoors portrait photography always think about what is the theme, what effect you are after and who you are shooting and why. That should give you a clear answer on where to shoot. The most important is to capture your model at their best, and choice of a location is just another tool like your camera or lenses.