This week's book giveaway is in the OCAJP forum. We're giving away four copies of OCA Java SE 8 Programmer I Study Guide 1Z0-808 and have Jeanne Boyarsky & Scott Selikoff on-line! See this thread for details.
Looks like connector-g.jar contains a package ( com.mysql.jdbc.trace ) that is not in the other, not sure what it is for, possibly debugging connector source. Probably just use the other one, without the -g.
MySQL Connector/J is distributed as a .zip or .tar.gz archive containing the sources, the class files a class-file only "binary" .jar archive named "mysql-connector-java-[version]-bin.jar", and starting with Connector/J 3.1.8 a "debug" build of the driver in a file named "mysql-connector-java-[version]-bin-g.jar".