What is the difference between a student and a teacher account?

They are practically the same. Once you create a classroom with your Duolingo account, you will be able to track other Duolingo accounts, following their language learning progress and assign tasks for groups of Duolingo users (classrooms). If you create a Duolingo account through Duolingo for Schools home page, you will also automatically have a "regular" Duolingo account.

In more detail, you can access Duolingo for Schools right now and sign in as a "teacher" with the login information of any Duolingo account—meaning, you can sign in with your Duolingo username or email address. From the Duolingo for Schools dashboard you’ll be able to invite other Duolingo learners to join your classroom. The people you invite will be able to see your email address. 

A “student” account is the default account type at Duolingo. You can quickly create your own for free on the main Duolingo website if you want to know what students see. Teachers, when using Duolingo for Schools, can invite Duolingo learners to join their classroom. Once a student joins and gives permission for the teacher to track them, the teacher will be able to see all students side by side. They can see activity details (when a student practiced, for how long, how many points they earned, how long their practice streak is), as well as the name and email address associated with the account. A teacher can also view curriculum content, hide “mature” words in the courses and restrict access to social features (such as forums) for all their students. Important to keep in mind: teachers can reset student passwords, so a student should only join classrooms of teachers they know and trust.

When a teacher has more than one student in their classroom learning the same language, they can also activate classroom-exclusive lessons specially tailored for them, based on the average student level of that class.