I want to enable the ASP.NET MVC 4\'s SimpleMembership API to integrate with my own database schema. I have a plain and simple table in my database called Users
You could create your own membership provider by extending the MembershipProvider. See Custom MembershipProvider in .NET 4.0 for more details. Using this approach, you would only need to implement the methods you need. This should help keep things more simple and wouldn't require you to add tables you don't need.
The basic steps (taken from the linked SO answer) are: