The color temperature its it's measure of how the spectral output energy is distributed across the visible spectrum. The higher the number the deeper it goes into the blue, the lower the number the more red-weighted it is, which is a "warmer" color. Mid-day sunlight has a color temperature of about 5800K, which is pretty blue. The color spectrum of a 100 watt halogen incandescent is about 3000-3300K. A 60 watt incandescent comes in about 2800K. Most "warm white" LEDs & CFLs come in between 2800-3000K.
But at the same color temperature, the quality of that light can differ, and is highly dependent upon it's CRI, but even that isn't the whole story.
CRI is a rough measure of how even or smooth that spectral output is. A CRI of 100 would mimic the spectral smoothness of natural light. A CRI of 60 would mean that there are "spikes" of single color light with deep valleys between them, and very little fill. Typical $2 twisty CFLs might be in the 78-82 range, but there are versions hitting in the high-80s as well, which is pretty good. (Incandescent 60 watt bulbs run a CRI of about 90.) This is due to the nature of how that light is generated with fluorescent technology. Fluorescent bulbs run an electrical current through a low-pressure mixture of argon gas with a hint of mercury (to bring the arc strike voltage down to reasonable levels). The visible light output of an argon tube is a single color line (blue), centered on the quantum energy levels of the outer electron shell (if you skipped high school chem & physics don't sweat it.) There is also a fairly strong line in the near-infra-red (not visible). The glass of the tube is coated with some magic-mouse-milk paint called "phosphor" that absorbs much of that light energy then re-emits the energy at a different color.
The mix of colors coming out approximate a white light, but it typically has three very strong color lines, one in the blue region, one in the green/yellow range, and another in the red. The exact mixture of the phosphor is what balances those lines to be either a "daylight" 5800K bluish light vs. a 4200K "cool white" vs, a 2800K "warm white". But the amount of "fill" color between the strong lines is limited.
LED assemblies work differently. Most better LEDs start with multiple high-efficiency LED elements, which can come in red, amber, yellow, green, and blue, all of which have much wider spectral peaks, not spikes. Sometimes a bit of phosphor is added to the blue elements to fill in better, but not always. The color temperature is tuned by the amount of power delivered to each LED die type. The highest CRI LED assemblies there will be at least three different LED types with fewer gaps (valleys) in the spectrum, and can hit north of 90 for CRI. Cheaper LEDs may only have one or two LED types and fill in with phosphors, but the CRI of those may be quite a bit lower.
A Cree TW /True White has at least two (possibly three) LED types, and runs a CRI of 93 which is pretty good. I'm not aware of CFLs that test that high, though some linear T8 tubes do. The only reason it isn't higher still is that it is enclosed in a glass that absorbs a bit of the yellow. It's spectrum can be seen here. The CRI of incandescents also suffer a bit from the absorption spectrum of the glass, as well as other factors.
So, you're probably not crazy- the quality of the light IS different. You are perceiving it as "whiter", but there isn't a spectral definition of "true white". The solar spectrum is sort of the gold standard, but even it has absorption lines from the atmospheric gases (predominantly those of nitrogen), but peaks well toward the blue end of the visible region.