Yes, the pound sign is different than the sharp symbol, just as single quotes are different than apostrophes, and guillemets are different than less-than and greater-than symbols, and the flat symbol is different than "b".

Now, limitations of first common US typewriters and then the ASCII character set have led to conventions where some of these have been used in place of the other because the correct symbol wasn't available, but that's different than the symbols being the same, and in most modern contexts, the correct symbol is usually available.