It really depends on the
theatre, as ever theatre is different.
In out theatre, the Production Manager, as gaff said, oversees all of the production side of things. He is the one who makes sure all of the shops (costumes, scenery,
props, lighting) have the staff they need and the information they need to get the shows produced. He is the person that the department heads report to.
Out T.D. oversees the scene shop, painters and the props shop. I, the ME, run my own department. The T.D. as gaff said takes the designer's drawings and figures out who to build them. Then I come in and start deconstructing the set so that I can get lights on it... We barter over the lineset schedule, but the
TD usually wins. The TD is in charge of ordering the supplied needed for the shop, keeping track of tools and making sure things get serviced and such.
The Production manager position is usually a less hands on position and very much a management position. They are usually in charge of the budgets and hiring, and the business end of production. The TD position is a very hands on position, and is generally a more carpentry oriented position.