Is there a difference between "Speciality" and "Specialty"?
Solution 1:
There seems to be divergence on that point: See Wikipedia. Others seem to think so as well.
I suspect that specialty is American English and speciality is British English.
Solution 2:
Specialty is the word used in American English (the OED reports it's chiefly Northern American), while speciality is used in British English.
In medicine (as for what reported by the OED) both specialty and speciality are used.
Solution 3:
It's standard US medical editing practice to change instances of "specialities" to "specialties." Note the certifying agency: American Board of Medical Specialties.
Solution 4:
There's a good explanation of the difference here.
In addition to the regional differences among English-speaking nations, it points out that 'specialty' is more commonly used as an adjective; 'speciality' as a noun.
As a native speaker of British English, this is the distinction I make.