Producing light is complicated. I wrote about the alphabet soup of power and speed in my last post, but there's also color. After you figure out color temperature and how to use it to your advantage, you might find that knowledge is wasted because your CRI, or Color Rendering Index, is low. Your "daylight" lamp or strobe may be producing 5600K light that looks like 4000K in camera because temperature isn't the only definer of color.
The reason isn't so simple. Remembering that white light is actually a mixture of colors, it's possible to have the right mix of but each color is being produced unevenly. Where Green is peaking, Red might be dipping. Green and Red make Yellow, so there may be shades of yellow that are washed out or off-color. CRI is measured on a hundred point scale, with 100 being the perfect rendition of color. Your average fluorescent light may have a CRI between 60 and 70, producing very uneven light.
This is a real simplification of CRI, but to be honest I haven't found more complicated answers to be helpful. When purchasing a continuous light source, you'll want it to produce daylight, 5600K, and have a CRI of at least 85. Anything lower and I'd get squeamish.