Well as far as the "mental illness" argument goes, that's pure horse hockey! Here are some scientific studies. The problem has been psychologists like to theorize about people's mental states. They often have no idea what real scientific research has done in the medical field.
General Male Female Brain Differences
Gorski, Gordon, Shrayne, Southham 1978
Medial proptic nucleas (MPON)
Allen, Hines, Shrayne, Gorski 1989
Hypothalamus regions
INAH3 2.8 times larger in men
INAH2 2 times larger in men
Regions known to influence:
gonotropin secretion,
maternal behavior in females,
sexual behavior in mammals
Allen 1989: associated area with sexual functions
LeVay 1991: associated area with sexual orientation
Allen, Gorski 1991
Shape of splenium of corpus callosum
Massa intermedia 53% larger in women
Anterior commissure 12% larger in women
Hoffman, Swaab
Preoptic anterior region of hypothalamus different between male and female
volume,
number cells,
neural structure
Transgender Involved or Related Studies
Garcia, Falgueras, Swaab 2008 hypothalamus
Region INAH3 volume and number of neurons in trans person matches the identified gender.
Zhou, Hoffman, Gooren, Swaab 1995 hypothalamus
Region BSTc volume and number cells in trans person matches the identified gender
Kruijver 2000 hypothalamus
Region BSTc number neurons in trans person matches the identified gender
Yokota, Kawamura, Kameya 2005
Corpus callosum: claim to be able to make clinical diagnosis based on size
Rametti, Carrillo, Gomez-Gil, Junque 2011
White Matter microstructures in transwomen similar to cis women
Simon, Kozak, Simon 2013
Grey Matter cerebellum, left angular gyrus, left inferior parietal lobe match identified gender
Zubiarre, Elorza 2013
Transwomen: feminization regions of right hemisphere
Transmen: substantial subcortal masculinization
Luders 2012
Cortical thickness of transwomen similar to ciswomen and greater than cismen.
Left hemisphere: frontal, orbitao-frontal lobe, central sulcus, perisylvian region, paracental cortex
Right hemisphere: pre/post central gyrus, parietal cortex, temporal cortex, precuneus, fusiform
Lingual, orbito-frontal gyrus
Other studies:
Govier, Diamond, Wolowiec, Slade 2010
Hearing response to particular sounds, trans people reacted similar to identified gender
Berglund, Lindstrom, Dheine-Helmy, Savic 2008
Smell: response to a particular scent, trans people reacted similar to identified gender (active brain scan)
Autoszewski, Zadzinski, Foczpanski 2009
Teeth: transgender tooth pattern difference (has something to do with the front teeth but one study of dentists could only identify male, female teeth accurately 50% of the time. The other 50% of the samples were ambiguous and no correct ID trend was found)
Behavioral Studies:
Bao, Swaab 2011
Showed rearing could not override inborn gender identity
Mathews, Fane, Conway, Brook, Hines 2009
Nordenstrm, Servin, Bohlin, Larsson, Wedell 2002
Both studies found prenatal androgen levels significantly effects play behavior and personality characteristics