An internship is a kind of position or job. An intern is a person who holds an internship works in that position.
An internship is a job that is intended to be a mixture of both paid work and also training or education. An intern is the person who does that job. An "intern" holds an "internship."
A very common use of the word is in medicine. An intern is a person who "is a doctor" in the sense of having an MD degree, but has not yet completed their clinical training and works under supervision.