Fluorescents vary in both their color temperature and color rendering index. The cheap industrial kind are often greenish and have a pathetic CRI. Many colors will just fade out to gray when viewed in this light. High CRI fluorescent bulbs are available in a variety of color temperatures, but they're seldom used in most settings - they cost more.
Incandescent bulbs all have good CRI (unless they're tinted), but their color temperature varies. Lower wattages tend to be yellower. As you said, your eye automatically adjusts for color temperature. Cameras don't. Fortunately, color temperature is relatively easy to change in post-processing - unlike a bad CRI, which isn't (your eyes can't compensate for it, either).
"Daylight" bulbs are often neodymium filtered. It makes the light look whiter but it does so by suppressing yellow - the apparent higher color temperature is bought at the expense of CRI. If you control the lighting and want light more like mid-day sun, don't use neodymium bulbs; a quartz-halogen will do better.
(Pet peeve: "flourescent" lights, which I guess are found in bakeries)