They're not the same. "They don't look anything like..." means "They look nothing like...".

Both are correct and mean exactly the same thing. However, depending on how stressed the word "anything" is when spoken, it can emphasise your statement.

The do have the same meaning. The word anything can emphasise just how different the two people. For example John has short blonde hair and pale skin. Jim has long black hair and tan skin. You could say they don't look anything alike because they are very different. Its just a stronger statement to show how different they are.