Adding a User to an existing SharePoint Group

I am trying to create a script to add a user to an existing SharePoint group. I'm new to PowerShell Scripting & SharePoint, so I'm a little lost. I keep getting the error that the specified group does not exist, even though I'm looking at the created group under the Site Settings -> Groups page.

What am I doing wrong?

#Add SharePoint PowerShell SnapIn if not already added
 if ((Get-PSSnapin "Microsoft.SharePoint.PowerShell" -ErrorAction SilentlyContinue) -eq $null) {
 Add-PSSnapin "Microsoft.SharePoint.PowerShell"
 }

Write-Output "Starting process"

$site = Get-SPSite 'http://spdev:36600'
$user = Get-SPUser -web $web -Identity DOMAIN\username
$rootWebGroups = $site.RootWeb.groups

Write-Output "Site: $site"
Write-Output "User: $user"

foreach ($web in $websites){
    $group = $web.sitegroups['PowerShell AddUser Test Group']

    Write-Output "Current website: $web"
    Write-Output "Group:: $group"
    Write-Output ""
    try {
        Set-SPUser -Identity $user -Web $web -Group $group   
        Write-Output "$user added to $group"   
    } catch {
        Write-Output "$user could not be added to $group"
    }
}


Write-Output "Finishing up"

And the error I am getting:

Set-SPUser : The specified group does not exist.
At line:27 char:19
+         Set-SPUser <<<<  -Identity $user -Web $web -Group $group.Name
    + CategoryInfo          : InvalidArgument: (:) [Set-SPUser], PSArgumentNullException
    + FullyQualifiedErrorId : Microsoft.SharePoint.PowerShell.SPCmdletSetUser

I did find a way that works by using $web.sitegroups['PowerShell AddUser Test Group'].AddUser($user) instead of #Set-SPUser -Identity $user -Web $web -Group $group, but I still don't understand why the previous code did not work.