They're not the same thing, just very closely related. In practice, unless you want to specify a particular time to shutdown or to force an immediate unclean reboot/halt/poweroff, it really doesn't matter whether you run
shutdown -h or
shutdown -r vs
reboot. Things weren't so nicely convenient in the past, but this is the way it works now (a lot of the opportunities for ambiguity or user error have been removed/smoothed out).
/sbin/shutdown does a lot of "cleanup" stuff like notifying users, blocking new logins while the system is shutting down. It can also be told to shutdown the system at a certain time or in XX minutes - warning logged-in users every so often of the impending shutdown.
It also changes runlevel to
reboot) which triggers the system to stop the running services, unmount disks, etc in preparation for the actual halt, poweroff, or reboot.
When all that is done, it then calls
poweroff are usually symlinks to
halt, which interprets them as
halt -r and
halt -P respectively).
Note, however, that if
poweroff is called when the system is NOT in runlevel
6 then they will call
shutdown to do its job unless the
force option is used.
(The details of how run-levels are interpreted may vary from distro to distro, but
6 are used as described here in Debian and Debian-derivatives like Ubuntu).
See the man pages for
halt for more info.
Finally, the distinction between
poweroff state is that
halt does everything up to actually powering the machine off (on some kinds of hardware this means it drops into a ROM bootloader or similar. On most PC hardware it just halts), while
poweroff completes that final step and switches off.
reboot, of course, gets to the halt state and then reboots the computer.
BTW, the default is typically to either
reboot, but you can use
-H as an argument to
shutdown to make it halt instead. This isn't very useful on PC-based linux boxes (but is useful on machines, like sparc boxes, that have a firmware boot monitor. It's possible it may become useful on newer machines with an EFI command line).