(I'm a U.S. native speaker). I don't believe there's any difference at all. "Theatre" or "Theater" is by far the most common, ordinary term nowadays.

Checking a dictionary, it simply gives "theater" as the first definition for "playhouse," with no notes or explanation.

Wikipedia says that "Playhouse is a common Elizabethan term for a theatre, especially those built in London such as The Globe and The Rose." That would explain why "playhouse" is used when discussing Elizabethan theater.

A quick Google search of Shakespeare's works at shakespeare.mit.edu shows that Shakespeare himself used both "theatre" and "playhouse," a couple of times each.