Yes, insomniacs have different brains.
More Info: A study conducted by Dutch researcher Ellemarije Altena , University of Cambridge, concluded that brains of older adults who suffer from chronic sleep disorders look different and function differently than adults free of sleep issues.
The researcher set out to analyze the differences in brain function and appearance in those suffering from chronic sleep disorders versus those without sleep disorders.
The study recruited a large number of participants who suffered from sleep insomnia to carry out a number of tasks while brain activity was simultaneously recorded.
Surprisingly, the study participants that suffered from chronic sleep disorders fared better on small cognitive tasks such as response time when given a visual cue. The researcher studied the brain activity during the tests and found that those with sleep problems have lower brain activity though they performed better. She speculated that this may be due to a form of hyper arousal wherein the brain makes use of other mechanisms to compensate for the lack of sleep.
The study also concluded that the brain of those with sleep disorders has a different appearance. The grey matter present in the brain was less dense in certain areas for those with sleep problems.
“Insomniacs Have Different Brains.” Netherlands Organization for Scientific Research. N.p., 16 Mar. 2010. Web. 24 Mar. 2011. <http://www.nwo.nl/nwohome.nsf/pages/NWOP_83LJ4S_Eng>.