The usual reasons for doing a portrait in a photographer's studio include control over lighting (light is paramount in all photography, whether serendipitous or bent to the shooter's will), control over weather (provided artificial climate control is working!), minimizing distractions, and availability of backdrops. In other words, control, control, control!
It is not unusual to bring artificial light sources and modifiers to a portrait location (which in this context simply means anyplace outside of a photographic studio). In many cases, this can be the best approach to meeting the requirements of a shoot. It does require a lot of time and preparation, however.
As an example of a "third way" that digital photography facilitates, here is an example where a company provided me with an image of a background of a sort that they now use for head shots. I did the shoot in my studio, against a green screen. If the screen is lit evenly and not permitted to bleed color into the edges of a portrait subject, it can be surgically removed in Photoshop, and replaced with any background available.
My subject could have appeared in front of the Eiffel Tower or International Space Station, but the goal was simply a very natural appearance, with a little more interest than a plain studio drape. As always, my primary aim was a relaxed, friendly, pleasant, competent and professional look. Would you hire this man? I would.