Mysqli_connect is the newer version of mysql library.
Here I in mysqli stands for improved.
Few things have been introduced with Mysqli.
They are,
-Prepared statements.
-Object oriented interface.
-Support for multiple statements.
-Embedded server support.
The mysqli extension has a number of benefits, the key enhancements over the mysql extension being:
-Object-oriented interface
-Support for Prepared Statements
-Support for Multiple Statements
-Support for Transactions
-Enhanced debugging capabilities
-Embedded server support
So Mysql_connect() basically is the database connector to Mysql whereas Mysqli_connect() is the connector for Mysqli databse