Þorn was in use since Fuþark (Germanic runes) but wasn't used to write Anglo-Saxon until around the 8th century. It died out after the printing press came into use, usually imported from France (or Germany or something occasionally) and not using some characters found in English at the time. Because of the lack of a Þ/þ key, typers started to use "Y" as a substitute (which is why you see e.g. "ye olde" instead of "the olde"). Eventually þorn just disappeared and people used the spellings using "th". A similar thing happened to Yogh (Ȝ/ȝ), where it was substituted for by "Z" (With e.g. "MacKenȝie" yielding "MacKenzie" instead of "MacKenyie") until it disappeared and spellings using "y"/"gh" (or "j"/"ch" when appropriate) replaced spellings using "ȝ".
Ðæt (Ð/ð/đ) was mostly replaced by þorn by Middle English so it didn't get to be slain by the printing press. Wynn (Ƿ/ƿ) was replaced by "uu"/"w"/"u" by Middle English too. Ash (Æ/æ) didn't die off, in large part because it was available on many printing presses of the time due to its usage in French and Latin, but it became obsolete for English words and was mostly used to replace "ae" in loanwords (especially from Latin and Greek).
There were some other funny things in Old English & Middle English orthography; like omitting n/m and writing a macron over the preceding vowel to indicate the sound (like "cā" instead of "can"), in the same way that it occured in Latin/Latinate languages which lead to "ñ" and "ã"/"õ" in Spanish/Portuguese/Galician.