Why are some state executive departments different from those in the federal government?

Different needs and powers in the states than in the federal government. 

Being states can create specific laws for the state there can be more specific departments established in the state government. Additionally, the states have more power to create laws for the local level than the federal government. This power comes from the 10th Amendment in the Constitution. Each state sets their own Constitution and from there departments needed for the state to run effectively.