What is the difference between state board for nursing and the nclex?

The NCLEX (National Councel Licensing EXamintation) is the test given by the group given the authority to create a standardized test for examination of potential nurses.

The state board of nursing is a group particular to each state who has the authority to grant or deny a license to practice nursing in a given state, among other things.

Basically, the NCLEX is the test. The State Board of Nursing will decide whether you are fit to be licensed as a nurse using those test results and any other criteria such as your educational program, criminal background check, etc.