Theater presents experiences through actors and includes the performance of the arts including plays. A play includes written dialogue between characters for the purposes of theatrical performance and may represent real or imagined occurrences. Drama is mostly associated with imagined performances it is exemplified by the sad and happy face symbols. These symbols point to comedy and tragedy which can be presented through drama. Drama may also refer to plays that don’t fall in the two categories of comedy and tragedy. In a nutshell theater includes the performance of plays and drama. Drama can be used as a term that refers to a type of play that is neither sad nor happy. Another distinction between plays and drama is that plays are meant to be performed while drama may be written for reading purposes.
Let's stick to the definitions from The British Theatre Guide and then part from there.
According to TBTG, drama is a branch from theater based on a representation of an event written in prose or pantomime that is meant to duly enact and illustrate an event or situational conflict through acting.
A play is the conduit upon which the event is represented. It consists on stage directions, dialogue, prologue or preface, author's notes and even the dramatis personae who will be included in the drama. Drama would be the opposite of Comedy, which is also a lifelike representation that views conflict under a lighter, jovial and paradoxical light.
The theater is the scenario or the stage on which the play drives the drama. The drama contained in the play unveils and expands on stage. Notice that even in the military the term "theater" is widely used as
An area defined by the geographical commander or Department of Defense as the place where a specific operation will take place.
Circa 1660, the meaning for theater was "place of action" as well as "region of war" therefore, special attention should be given to the meaning of "theater" as an actual place or location, and not just as the meaning of the art itself. "Theatrical Arts" would be the correct usage for the word "theater" as a branch of humanities.
Drama and a play are one and the same. They are the printed text of the play (verse or prose). Drama can also be defined as the plays which are produced for the theatre structure. Including: the stage direction, lines, and character list. Theater is the actual production of a play. The visual representation of the play. The theater, and the stage in particular, is the place that personifies the characters. Theater can also be defined as the place where a play takes place. It is the personification of the characters. For example, Shakespeare's Globe is a theater where a play is produced and shown to the public.