What's the difference between a user's home path and their profile path in Windows Server 2003?
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.)
HOMEDRIVE/HOMEPATH is where the user has his personal files, e.g. downloads, music, documents, etc.
HOMESHARE is used instead of HOMEDRIVE if the home directory uses UNC paths.
USERPROFILE is used to store the user's application and OS configuration files and personalization settings. It includes both local and roaming (Active Directory) folders.
It's important to note that although HOMEDRIVE/HOMEPATH is often the same path as USERPROFILE, it's not always the case.
I can answer for Server 2003 AD ... there may be some adjustments in Server 2008 AD.
1- The "profile path" is the location where profile specific information is stored. By default, it is a location on the specific computer (see Kuba's answer). When a path is filled into the AD value for the user, a roaming profile for that user is activated, and the profile is stored at the network path specified, and copied/updated at that location whenever the user logs in.
2- I don't know how the "local path" setting in the "Home Folder" section functions, as I have only seen that section left default, or the "Connect" setting used to map a drive letter to a network path. At my current company, we do neither, using a Kixstart login script to map a drive to a "home" folder.