I'm pretty familiar with the Clannad anime, and I can tell you with 99.9999% confidence that this song never appears in it.
As I said in my comment, and as Toshinou-san and senshin reaffirmed in theirs, it would be very unlikely for two anime to use the same song, especially if one of them is a Studio Ghibli production. Not impossible, but unlikely. As in "I've been watching anime for seventeen years and I've never seen two of them use the same song". That kind of unlikely.
However, that still doesn't mean "impossible", so I looked into the song, "Inochi no Kioku", to try and find out more about its origins. It was written and performed by Japanese singer Kazumi Nikaido. An Anime News Network announcement about the Kaguya Hime film mentions that the song was chosen as the movie's main theme.
The wording of the announcement makes it sound as if Nikaido created the song independently of the film's production, and the production staff decided to license it for use in the film. This is also supported by the release dates; the single for "Inochi no Kioku" was released 24 July 2013, according to GhibliWiki, while the film was released 23 November 2013 according to Anime News Network.
Licensing a song like this is somewhat unusual for anime; it's more typical for the production to hire a songwriter to write a custom song for the anime, and hire a singer (often one of the voice actors) to perform it. They also sometimes hire singer/songwriters to both write and perform the music. I can only think of a few cases (e.g. FLCL with The Pillows) where an anime production licensed music which had already been released.
However, the Anime News Network page on Nikaido only lists her as staff on one anime, The Tale of Princess Kaguya. This strongly suggests that "Inochi no Kioku" only appeared in that anime. I think your memory might be playing tricks on you :)