pconnect will open a persistent connection. If one is already open using the same user/pass identifier, it will use that one, and thus you would (should) only have 1 connection open at a time.
connect will open a connection for the length of execution of the script (or until mysql_close() is called).
Note: Use of the MySQL library is now considered discouraged. Consider using the mysqli or PDO libraries.
The number of opened connections to a mysql server can be found by issuing a query for SHOW STATUS; this will, among other things, return a field for Threads_connected