Anything: More general. (A waiter asks an open-ended question: "Do you need anything?")
Something: A specific thing. (A friend asks: "Can I get you something to drink?")
In certain situations, people may use "anything else" when they are being sarcastic or are frustrated. For example:
-Annie does not want the food her mom cooked for dinner, so she asks, "Mom, can I have something else to eat?" (This request sounds more polite).
-Annie does not want the food her mom cooked for dinner, so she says, "Mom, can I have anything else to eat?" (This sounds rude because she is asking for anything BUT the food provided).