They look the same but they are not the same. The meaning is different.
Archaic = the very first time (example: Archaic Period)
Archaeo = very old (example: Archaeology)
arche + type = proto + type = the very first type ( the very first that we know )
Now, when you see the phrase in the parenthesis which is : “the very first that we know”, something changed, because you suspect that what we know might not be the very first type. And you’ re right. Despite the fact that the meaning of the word is literally : the very first type.
This is because the scientists, or the researchers, like to use words, things and stuff in order to organize their thoughts or to create their own definitions of things that they have on their mind, which is -sometimes- very far away from the meaning of the word that they use as a title.
Here’s an example : When the archaeologists had the very first discoveries, they thought that the discoveries belong to the very first period of the human civilization, and they thought it would be a great idea to give the name ‘Archaic’ for this period of time. Actually they were very wrong, very wrong. And they still use the same term, despite the fact that they already know they were wrong.
Same shit, you’ ll meet in psychology and in psychiatry. And not only. This is one of the reasons you get confused most of the times.