Java and .NET are 2 different programming languages. One is created by Microsoft and one is created by Sun Microsystems.
As far as DB connectivity. Think of ADO.NET/JDBC/ODBC as methods to connect to databases. The reason you would use ADO.NET over JDBC is simple... JDBC won't work with .NET applications hence the name JDBC(Java DataBase Connector). ODBC is the Open DataBase Connector and can be used with alot of different languages but often isn't.
The two are similar but are losing popularity to PHP which is going to be the most popular language in the world within the next 2-3 years.
Hope I answered your question.
Take care!
Brian