The tasks categorization of overdue & due today is showing different in app & in website

Yes, this is a default feature. When you assign a task to someone, a mail notification is automatically triggered.

However, in your case, the email configuration seems incorrect, which is why the email is not being sent. Please verify the password in the email settings, save the changes, and try again.

You can also refer this post for email configuration :

Let us know if you are still facing the issue.