Continue execution on Exception

You use try {...} catch {...} when you want to handle errors. If you want to ignore them, you should set $ErrorActionPreference = "Continue" (or "SilentlyContinue") as @C.B. suggested, or use -ErrorAction "SilentlyContinue" for the particular operation raising the error. If you want to handle errors from a certain instruction, you'd put that instruction in the try {...} catch {...} block, not the entire loop, e.g.:

foreach($user in $users) {
  ...
  try {
    if(($userlogin -like "$oldprovider*") -and $convert) {  
      LogWrite ("Migrating User old : " + $user + " New user : " + $newalias + "    ")
      move-spuser -identity $user -newalias $newalias -ignoresid -Confirm:$false
      LogWrite ("Done")
    }   
  } catch {
    LogWrite ("Caught the exception")
    LogWrite ($Error[0].Exception)
  }
}