I have a whole area that I setup with CFL's. I pack them pretty tight. Here's the spectral curve on a 6500K natural daylight bulb. The curves are significantly different than a Tube Fluorescent. The CFL's have a pretty broad color temperature and a spike at 6500K. I was very surprised when I ran this curve. It is more of a full spectrum bulb.
The warm white is an entirely different animal. It had a color temperature of 5000K but had a significantly narrower color range unlike the natural daylight bulb. Most fluorescent bulbs with 5000K are considered "Neutral" instead of Warm White.