According to various veterans interviewed at the time the movie came out, it appears to be very close to reality. Many veterans said that they couldn't watch the first 30 minutes of the movie because they couldn't stomach going through it again. Spielberg hired a number of outsiders and WWII historians to make the entire movie feel as realistic as possible, such as putting the actors through a 10-day boot camp. I'll link some of the interviews, and a transcript of a PBS piece on Saving Private Ryan from when the movie came out if anyone wants to dig further into it.

http://articles.philly.com/1998-08-06/news/25724660_1_omaha-beach-va-center-nearest-va-facility

http://www.pbs.org/newshour/bb/entertainment-july-dec98-ryan_8-3/

http://jacksonville.com/tu-online/stories/081598/met_2a1priva.html

Despite common beliefe the opening szene is very exaggerated. The sector where Hanks lands would be pretty much a death trap for everybody going a shore within a short amount of time. If you watch the szene you see at least 4 Mg-42 in a section of 100 m with good vision on the beach. The Bunker in the film and the pillboxes are far closer to the beach than in realitiy also higher elevated. Like i said there is no way that you can land soldiers so close to such massiv amount of enemy fire. The film also, and i hope i don't get too much flak for this, avoids showing that a lot of american soldiers landing at that beach tryed to hide and didn't move forward as planned( not trying to diminish their courage i would have shat my pants and vomited until i lose consciousness).

It took them very long to actually take the beach against a very limited number of defenders who were farther away than showed in the film.

In short: the film exaggerated the german defense and the speed of the american advance.

The advance over a longer beach with pillboxes hardly visible at the end of the beach would have not looked so impressive.