The reason I say partially is that my experience with lucid dreams is that sometimes I'm aware that I'm dreaming. And sometimes I can control what I'm doing. But other times I'm in control and conscious but I'm not quite at that point of realizing I'm in a dream. It's possible those are full lucid states, but I simply wasn't paying attention to the signs of unreality as I was focused on something else.
A recent New Scientist article  points out publications last year (primarily of Ursula Voss and Allan Hobson) that try to find neurological correlations of lucid dreaming. Furthermore, these researchers want to use lucid dreaming to explore a primary-secondary theory of consciousness. As Jessica Hamzelou explains in the New Scientist article:
Gerald Edelman at the Scripps Research Institute in La Jolla, California, proposed that there are two possible states of consciousness, which he called primary and secondary consciousness. Primary consciousness is the simple subjective experience of sensory perception and emotions, which could be applied to most animals. It's a state of "just being, feeling, floating", according to Ursula Voss at the University of Frankfurt in Germany.So, on the premise that there is secondary and primary consciousness, lucid dreaming may be dreaming with secondary consciousness active, whereas non-lucid dreaming involves only primary consciousness.
The mental life of your common or garden human, however, is a lot more complicated. That's because we are "aware of being aware". This allows us to reflect upon ourselves and our feelings and, in an ideal world, make insightful decisions and judgments. This state, dubbed secondary consciousness, is thought to be unique to humans.
The study of Ursula Voss et al  conducted at Frankfurt University found differences in the brain patterns (measured with EEG and CSD (current source densities)) between lucid sleep, REM sleep, and waking with eyes closed. It's only a small amount of data, but it helps show that lucid sleep is not just a part of REM sleep, but shares correlations with waking.
Some other info that may be of interest to those who want to conduct lucid experiments: Only half of the subjects could enter lucid states in the laboratory, even though all of them claimed to have lucid dreams beforehand in non-laboratory settings. The researchers were not able to induce lucid dreaming in the subjects with machines--the only reliable way was for the subjects to self-induce lucidity. The duration of lucid states was ambiguous. This experiment ended up with only three recorded lucid episodes. As the report admits, there are still major methodological issues.
Due to the types of measurement, Voss was not able to test the hypothesis that the dorsolateral prefrontal cortex (DLPfC) would light up in lucid dreaming, as compared to REM sleep in which the DLPfC takes a break. If DLPfC is a site of "executive ego" than it should be active in dreaming when one is lucid but not otherwise and may correlate with Edelman's secondary consciousness.
Allan Hobson of Harvard Medical School, another researcher mentioned in the New Scientist article, wrote a paper last year about the pros and cons of using lucid dreaming as a tool for investigating the neuroscience of consciousness . He references the Voss experiment, and like Voss points out the untested hypothesis that the DLPfC should activate in dream states that are lucid. Hobson also references studies led by Michael Czisch that used MRI to measure differences in lucid dreaming (according to Hobson). The imaging showed that certain frontal, temporal, and occipital regions have increased activation during lucid sleep--regions distinctly human as compared to macaque monkeys. Since those regions might be key for consciousness research, we might have a way to test aspects of consciousness built into our brains that we have hardly exploited.
A commentary of Hobson's paper by Don Kuiken of the Department of Psychology, University of Alberta, was recently published . A possibility Kuiken points out is that perhaps in non-lucid states the brain uses an alternate form of self-regulation similar to that in musical improvisation. It is interesting that the brain might be using the same patterns when playing jazz as when in REM sleep--going with the flow. So, as Kuiken offers, non-lucid sleep regulation and "responsiveness to 'what comes'" might also be important for consciousness research.
 Hamzelou, J. "Want to find your mind? Learn to direct your dreams," New Scientist, no.2764, 15 June 2010.
 Voss U, Holzmann R, Tuin I, Hobson A. "Lucid dreaming: a state of consciousness with features of both waking and non-lucid dreaming," SLEEP, vol.32, no.9, pp.1191-1200, 2009.
 Hobson, J.A. "The neurobiology of consciousness: Lucid dreaming wakes up," International Journal of Dream Research, vol.2, no.2, pp.41-44, 2009.
 Kuiken, D. "Primary and secondary consciousness during dreaming: Commentary on 'The neurobiology of consciousness: Lucid dreaming wakes up' by J. Allan Hobson," International Journal of Dream Research, vol.3, no.1, pp.21-25, 2010.