What is the difference? What is mostly used overall?
If you wanted to invest in something inexpensive and safe for indoor use...and also outdoor.
If you were using it to shoot people (sounds terrible).
Merci.
Well. Huge differences. Technically speaking they are different kinds of instruments, creating light in different ways. We'll skip that and get to the parts that are of interest, use in film making.
"Tungsten" is sort of a generic term these days that can refer to any light in the 3200K color temperature range. If that's confusing, then I recommend looking over this:
http://en.wikipedia.org/wiki/Color_temperature
You'll notice that 3200 falls in the range of an "incandescent light bulb." Household incandescent live in this territory, but will vary. 3200K is what "pro" lights are balanced to create when shooting "tungsten."
I'll skip over daylight (5600K typically), but the wiki should cover that well. 3200K and 5600K are probably the most common light temperatures you will see when shooting.
Now, with fluorescent light things get a bit dodgy. Are you talking about CFL style lighting, or the sort of fluorescent light you would see in a typical office building? There are minor differences, but they all have varying levels of one particular problem:
Green Spike.
Standard fluorescent light does not give an even curve on a spectrometer. There are spikes at specific wavelengths that create our interpretation of the color of the light. Particularly green spikes the hardest. This is true even in "balanced" CFL bulbs which are supposed to give a specific color temperature. (2700K and 5000K are common now a days) Even though these bulbs have a measurable temperature output, they do spike specific wavelengths, and still spike strongly in green.
This, as you may have guessed looks TERRIBLE when lighting people. Regular office/household flourescents are the worst. There are gels specifically to remove the green from this sort of light. Interestingly, this is the use of the "tint" metadata feature of the RED camera. "Tint" in this instance is actually a magenta/green slider allowing you to adjust for this sort of thing when post processing the RAW footage.
Among all the schools of thought and the varying techniques for getting beautiful skin tones, the one thing that seems to be universal is that the green spike in flourescent light is just brutal.
If you want an example of what this sort of light looks like to a tungsten balanced camera, watch the drug store scene of Natural Born Killers.
That said...
There are lots of CFL lighting tools available these days - very interesting. All of the bulbs I tested recently still had a small amount of green spike to them. Not really enough to worry about gelling them, but if I were using those exclusively I would probably adjust out some of the green in one way or another.
Also something else problematic I ran into when looking at CFL solutions:
I could find "high output" bulbs (up to 300w equivalent), let's call that feature A.
I could find dimmable CFL bulbs (feature B)
I could find "balanced" CFL bulbs (Feature C)
I could not find all three (A, B, and C) in a single bulb. I could find A and C, or A and B, but not all of the above. The bulbs that were dimmable were useless from a color temperature perspective. The bulbs with a semi-standard color temperature (or at least consistent so I could predict their usage) were not available as dimmable high output bulbs. That got sort of frustrating.
Regarding safe for indoor/outdoor use - that is more a function of the equipment than the light bulb itself. Poke around at some of the "Chinese knock-off" lighting threads on this board. Darty's posts in those threads about what to look for in a lighting instrument are basically canon when it comes to safety.
Aegis mentions an important point that I forgot to add - we see things using eyes that constantly adapt to lighting conditions. Cameras cannot do this, especially with color temperature.