Basicall, one is an amended, expanded version of the other, both by Geoffrey Cornish & Ron Whitte.
"The Golf Course" is 1981, is 365 pages, lots of color. "The Architects of Golf," 1993, is 648 pages, all black and white imagery, and drops some earlier material on shot values and the evolution of golf course features but adds a lot more on Pete Dye and other modern designers (Desmond Muirhead, Rees and Bobby Jones Jr., Tom Fazio). Also has much expanded-profiles of architects and comprehensive lists of courses, plus some additional bibiliographic and resource help.
I prefer the earlier book for its simplicity and imagery, whereas "Architects of Golf" is much more inclusive. Book needs to be updated and reissued in another, totally revised third edition with color, but the work is enormous and a publisher is very reluctant since the first versions sold so amazingly well, they feel there might not be that much of a market left for a new version. But I think it would work.