That history actually isn't quite as clear. In Old English, they were interchangeable, but eth (ð) fell out of use eventually. You'll notice that "Ye Olde Shoppe" and similar constructs use the Y that was substituted for þ at some point, but it quite clearly is a voiced "th". Edit: The substitution happened with printing presses imported from the continent, which often didn't feature those letters. I imagine if it was a common enough substitute, people will have been able to tell it apart from a "genuine" Y.

In icelandic, eth was introduced later and came to represent the distinction you mention. In the International Phonetic Alphabet, [þ] and [ð] are also used as you describe them. But Old English wasn't quite as cut and dry.