I don't really know. The '68 would probably be like a reissue type, whereas the standard LP Custom is more modern?
Anyways, you might want to check out the Jackson Sweetones... really nice jazzy guitars, but a bit on the expensive side.
http://www.jacksonguitars.com/SweeTo...GES/STAT3.html
http://www.jacksonguitars.com/SweeTo...ES/STBT2H.html