They mean the same thing, but the usage is different.
兄 is the older term and is somewhat old-fashioned. In modern mandarin it is typically not used by itself, but it still appears in combination with other characters in words like
兄妹 (an older brother and younger sister)
老兄 (a familiar form of address between men)
Whereas 哥/哥哥 is often used by itself to directly address or refer to an older brother. It is also a component in other words like 表哥 and 哥们儿，but unlike 兄，哥/哥哥 is commonly used by itself.