With a gold membership, you'll get the ability to play online with other people, receive exclusive content, and generally have a more active role in communicating with other members of the Live Community. With a silver one, you can track achievements and use the marketplace (so yes, you can get classic games with a silver membership).
Some more info from Xbox live:
Xbox Live Gold Membership:
Play your Xbox 360 multiplayer games online with the premiere online gaming service. Use the brand new TrueSkillรขโยข Matchmaking system to play against opponents with similar skills, personalities, and gaming tastes. Give player feedback to rate your teammates and opponents on their sportsmanship, abilities, and conduct to influence matchmaking. Play select original Xboxรยฎ games online (the Xbox 360 Hard Drive is required). Get access to exclusive Gold Member content. Engage in video chat. Enjoy all the Xbox Live Silver features (below). Xbox Live Silver Membership:
Go online free of charge. Register a unique gamertag and set up your individual gamer card to keep track of your gaming accomplishments. Engage in one-on-one Voice Over Internet Protocol (VOIP) and chat with your friends via the Xbox 360 Headset. Access Xbox Live Marketplace, download and view demos, trailers, and purchase classic and casual games over Xbox Live Arcade using Microsoftรยฎ Points. Preview Xbox Live Gold during free weekend trials. Download auto updates and free downloads for your Xbox 360 games. Access Xbox Live Message Center where you can send and receive voice and text messages from other members. Compare gaming stats and achievements against other Xbox Live members. Use your friends list to see who's online.