Not quite.
There are actually 3 "families" of controls to choose from:
HTML Controls are standard HTML controls that have been a part of the HTML
language for years. These are not classes, they are tags and you can only
do client-side programming with them.
HTML Server Controls are the exact same as above but have: runat="server"
added to them. These controls add server side programmatic capabilities to
the otherwise standard HTML Control. Use these controls when migrating old
web applications to .NET applications.
Web Form Controls (a.k.a. Server Controls) are .NET Classes that expose a
rich set of properties, methods and events. They are programmable on the
client and/or the server. There are some that offer the same UI as HTML
Controls but have more features and there are some that offer a much richer
UI that would take quite a bit of programming if they were to be made in
standard HTML from scratch (i.e. Calendar Control, DataGrid Control,
Validation Controls). All of these controls render to the client as HTML
and/or CSS and/or JScript. Use these when rich UI's and/or rich
programmatic capabilities are needed.