Do "coterminous" and "conterminous" have exactly the same meaning?

Yes, conterminous and coterminous both mean "to share a boundary".

According to the entries for co- and con-, below, co- is an Anglicising of con-, which is possibly why Latin purists prefer con-

In this instance co- and con- both mean together or with.

Etymology of co-

in Latin, the form of com- in compounds with stems beginning in vowels and h- and gn- (see com-). Taken in English from 17c. as a living prefix meaning “together, mutually, in common,” and used promiscuously with native words and Latin-derived words not beginning with vowels, sometimes even with words already having it (e.g. co-conspiritor).

Etymology of con-

prefix meaning "together, with," sometimes merely intensive; the form of com- used in Latin before consonants except -b-, -p-, -l-, -m-, or -r-. In native English formations, co- tends to be used where Latin would use con- (e.g. costar).

"Con-" has the meaning of :

used with certain words to add a notion similar to those conveyed by with, together, or joint congenial, congregation, console, consonant, construct, converge, etc.

It has the same meaning as "co-", which means: