How the white and colored noise differ in time domain
White noise looks like the plot below on the left in the time domain. A key point is no matter how much you zoom into it, it will still generally look like this. Why? Because by definition every sample (let's assume a sampled system although this is not limited to that) is uncorrelated from the next in a white noise signal. The auto correlation is therefore an impulse.

If it happens to be a Gaussian White Noise process, then it will also have a magnitude distributed as shown in the histogram on the right, but these are two different characteristics, it can be White and not Gaussian, or Gaussian and not White.



Now for comparison, below is the same result for a "Colored Noise", and in this case, Gaussian distributed process.  Interestingly here the histogram of the distribution, given sufficient samples is identical, but the key point is as you zoom into the time domain plot on the left, we see the result of "memory" due to the bandwidth limiting that was done. Because of low pass filtering (in this case), the signal cannot transition from one sample to any other arbitrary location on the next sample; the location of the next sample depends on previous samples and is therefore constrained. (The more we low pass filter, the longer the effective memory depth). So there is more predictability on the next sample based on previous samples; samples are correlated to adjacent samples, and the auto-correlation function will no longer be an impulse but will be spread out with a width depending on how narrow the filtering is in frequency.



In the frequency domain, the power spectral density for the "Colored Noise" could look like the plot below; in particular this plot is the low pass filtered signal discussed above (but colored noise could be any selective band of frequencies, or even multiple bands). In contrast the white noise would have the same level of power spectral density for all frequencies. So in case this point wasn't obvious to all readers; white similar to white light that contains all frequencies vs colored with again the analogy to light in that individual colors are given by selective frequencies of light.