Saving Private Ryan is the better film, hands down. It has a more personal focus, rather than trying to show the entire big picture in one huge epic film. It's an epic in its own way, of course, but it's more of a film that the ordinary G.I. can relate to.
The sixties were still Hollywood's time for making epic films, films on a grand scale, and that's what The Longest Day is. It's Big Hollywood Studio Film Making in its heyday, but that wasn't always a good thing. The black and white photography does work in its favor, in my opinion. In the end, though, for me it's basically a good attempt that sort of turns into another big box office vehicle for the big movie stars of that period.
The Longest Day earned a lot of money (for back then) and it earned its accolades, but it looks somewhat dated to me now.