To place something is more general and unspecific. It refers to the action of putting something in a place. It can be literal and abstract.
She placed the silverware around the table. (literal)
She placed first in the one-mile race. (abstract)
To position something is more about orienting something. The action of positioning something is to put it in a better state. It can also be literal and abstract.
She positioned the magazines on the coffee table in a more organized fashion. (literal)
He positioned himself well to receive the promotion. (abstract)