TempData is session, so they’re not entirely different. However, the distinction is easy to understand, because TempData is for redirects, and redirects only. So when you set some message in TempData and then redirect, you are using TempData correctly.
However, using Session for any kind of security is extremely dangerous. Session and Membership are entirely separate in ASP.NET. You can “steal” sessions from other users, and yes, people do attack web sites this way. So if you want to selectively stop a post information based on whether a user is logged in, look at IsAuthenticated, and if you want to selectively show information based on what type of user is logged in, you use a Role provider. Because GETs can be cached, the only way to selectively allow access to an action in a GET is with AuthorizeAttribute.
Update In response to your edited question: You already have a good example of using TempData in your question, namely, returning a simple error message after a failed POST. In terms of what should be stored in Session (beyond “not much”), I just think of Session as a user-specific cache. Like the non-user-specific Cache, you should not put security-sensitive information there. But it’s a good place to stick stuff which is relatively expensive to look up. For example, our Site.Master has the user’s full name displayed on it. That is stored in a database, and we don’t want to do a database query for it for every page we serve. (An installation of our application is used in a single company, so a user’s full name is not considered “security-sensitive.”) So if you think of Session as a cache which varies by a cookie which the user has, you won’t be far wrong.