Where I live a student can move directly from college to grad school for nurse practitioner training. I know that there are some programs and/or states which require a year or 2 of experience. The key, however, is that the applicant must have passed NCLEX and be certified as an RN. It is common for nurses ro take time to study for NCLEX and then take the test after graduation from their undergrad college. More often this explains the lag.
But this happens with med school as well. The average age of first year medical students is now 24. A friend of mine sits on the admissions committee of a medical school and tells me that they prefer candidates who have done something else for a year or two after college. Another friend tells me that this she finds this also to be the case for schools of social work.
With regard to nurse practitioners, they can diagnose and prescribe, in many cases just as a doctor would. They can maintain an independent practice as well. This freedom is the real advantage of an NP certification. By contrast, a PA can only work under the supervision of a doctor. So, there is a different level of freedom and responsibility for NPs over PAs.