They are basically interchangeable, somewhere around 3rd edition it became adamantine from adamantite. For example, there 4 references in the 1st edition Dungeon Masters Guide (DMG) to adamantite (see below for one instance comparison, emphasis mine), and zero references to adamantine.
DMG 1st Edition - Daern's Instant Fortress description (Page 142):
[...] The adamantite walls of Daern's Instant Fortress are totally unaffected by normal weapons other than those of catapult type. [...]
DMG 2nd Edition - Daern's Instant Fortress description (Page 165):
[...] The adamantite walls of Daern's Instant Fortress are totally unaffected by normal weapons other than those of catapult type. [...]
DMG 3.5 Edition - Daern's Instant Fortress description (Page 254):
[...] The adamantine walls of Daern's Instant Fortress have 100 hit points and hardness 20. [...]
Drow magical items do gradually lose their abilities and integrity when exposed to sunlight, however.
One online wikia that I looked at makes the claim that it was changed around 2000, however when I look through the Book of Aritifacts, copyright 1993, it refers to a couple of items as adamantine, and no references to adamantite.
Axe of the Emperors - Book of Artifacts (Page 19) -
The Axe of the Emperors is a double-bitted, twohanded war axe. The head is of the brightest adamantine, forged so that it looks like a rising phoenix, the talons wrapping around the socket to clutch a sizeable ruby.
The BoA also refers to adamantine in creating items as well (Page 126):
If other materials are needed beyond the stick, the most obvious additions are ornamentations fashioned to it. Gems, beads, pearls, and bands of silver or gold are common. More uncommon components are feathers, exotic leather, adamantine wire, skeletal claws, and the like. These may sit in, dangle, rattle, wrap, or point; a wand can have more than just a stick, too.
So it appears that somewhere in the early 1990's it was changed from adamantite to adamantine.