While there are more similarities than differences between what a teacher sees and what a students sees when viewing the same Manhattan classroom, it's worthwhile to point out the differences.
A major difference is that the teacher will have a

button on the classroom's Main Menu. Clicking on the button leads to a menu containing a number of options for administering the classroom. (See Configuration)
Since only teachers can post messages to the Handouts/Notices, Lectures, Internet Resources, and Self-tests modules, the

button does not appear on students' pages within these modules.
As described in Manhattan's Copy and Paste Commands, Manhattan's "Copy and Paste" feature is available only to teachers. Students will not see either of the buttons:

anywhere within Manhattan.
Only teachers can delete messages. Students will not see this:

at the bottom of any messages. Also, only teachers have the ability to delete messages from the Chat transcripts, as described in Chat Transcripts
The teacher's view of the Assignments module is quite different from that of a student. See The Assignments Module for complete details. Also, in the Team/Teacher module, the teacher gets to see a list of all the teams, while students see only their team - see The Team/Teacher module.