Shameless, blatant plagiarism:
The profile path is the location of the user's user profile. The "Home" path may be the same, but it could be set to another location (via the user account properties).
The home path is a bit of a vestigial thing. It dates back to Windows NT, prior to the 'My Documents' directory. I believe the original intent was to provide a "Home Directory" similiar to Unix environments, but the user profile ended up (with the advent of "My Documents") being the default storage location for files (which led to the whole "redirect folders out of the user profile" functionality that came on after W2K).
"Folder Redirection" can use the legacy home path setting as the destination for redirecting the "My Documents" path. This can be handy if you have groups of users who need their "My Documents" path redirected to various server computers, as you can set a different home path on a user-for-user basis. (You can do the same thing w/ multiple group policy objects, or with a single folder redirection policy based on group membership, too.)