Inside the readme file of the ASP.NET Universal providers NuGet package is this quote
The SqlMembershipProvider, SqlRoleProvider, SqlProfileProvider c
the universal providers enable you to use asp.net membership system on sqlazure and also make the implementation sql agnostic. you can take the same app using universal providers and change the datasource to be sql server/sqlexpress/sqlce/localdb etc
The main difference, as far as I can tell, is that when you connect to Sql Azure you need to implement retry logic. The original providers won't retry when a connection error occurs and this will happen from time to time with SQL Azure.
Also, the original providers don't raise exceptions containing the full SQL error codes so if your database is being throttled you won't know which throttling rule is being applied. At least this is what the SQL Azure support team tell me - I'm currently in the process of upgrading for this reason.
A note of caution: the Universal Providers seem to use a different database schema to the original providers, so you will need to migrate membership data. If you are starting a new project it will be much easier to change these providers before you go live with real users!