I think that it is generally considered that in Icelandic ð is voiced and þ is not. However they say that in Old English there is no difference. It appears that þ came from the rune and was the first to be used. Later the Irish trained monks introduced ð in Anglian areas and it quickly became popular. Some writers preferred one and some the other but most used both interchangeably. Our language course only uses þ and never ð and my paper dictionary, Clark Hall, only uses ð and never þ.
The letter þ is called “þorn” and is sometimes used for the word “thorn” but ð never is. We call ð “eth” but the Anglo-Saxons called it “ðæt”. Occasionally ð was used for “ðæt” but þ never was. The crossed thorn was used for “ðæt” and, in fact, was far more common than ð.
Transcriptions tend to rewrite that in full so my copy of the Peterborough Chronicles replaces the “ð” in the manuscript with þet, the Anglian version of þæt.
In manuscripts I have seen oþ and oð, þa and ða, þæt and ðæt, þe and ðe, þes and ðes, þone and ðone and wiþ and wið.