Difference between 哥 and 哥哥
哥哥 is more likely to be used by kids. Similarly, in English, children call their parents Mummy and Daddy but when they grow up, they would call them Mum and Dad.

As an adult, you can still call your brother or male cousin 哥哥 and that is not a problem. In fact, many people use both 哥 and 哥哥. It just depends on which they feel like to use at the moment. Generally speaking, 哥 also shows that you are a little bit closer to each other. 

Chinese people sometimes call their best friend 哥哥, 姐姐, 弟弟, or 妹妹; Each of them can be said only by one character, such as 哥 or 姐. Besides their cousins and siblings, people used to use them on their best friends only.

These days, 哥 and 姐 have been widely used. For example, sales people may call their potential customers 哥 or 姐。