Can someone clarify which to use and which one corresponds to the future direction Microsoft is taking?
If you are migrating old Membership Provider to new one, ASP.NET Universal Providers will be the closest one (as well as easiest).
does this mean that ADO.NET code will not work?
In Universal Providers, table names doesn't have aspnet_ suffix which is great. However, if your existing project accesses those tables, then you'll need to modify your source code.
will Universal Providers only work with Entity Framework?
Universal Providers uses Entity Framework code first under the hood.
If this is the case, what is the best migration approach to take short of abandoning Azure completely?
As I stated, you can migrate old Membership Provider to new Universal Provider because table are one-to-one except aspnet_ suffix.