The main difference between the two is the services that will be given to the users. Xbox Live Silver is known to be free. You will be given a chance to check out the Xbox Store so that you can check out the different games and DLC. If you choose to upgrade to Xbox Gold, you will be given a chance to play two to four games a month for free. You can imagine how amazing this would be.
Take note that the number of games will depend on the type of XBox that you have. Xbox Gold will also give you the chance to play with other players online, which can be fun and exciting for you.