I'm not a huge fan of either of these movies. I don't own either one. I feel both of them are given a little too much credit at times - SPR by the "academy" type and Matrix by fanboys and such.
BUT....
I have to basically repeat what I said before about them. Saving Private Ryan to me has a story that doesn't match up to its film making. The best part about it is the opening sequence and then it goes downhill. If you had any other director doing the exact same film script, nobody would know it even existed similar to other war movies like uh...Hart's War? Is that what it was called? The only other thing that this has over Matrix is acting.
Now, the Matrix is overrated in its slow motion technique. Its not as amazing as people make it out to be. But the style of the film outside of that holds up as well. And on top of all that, it has a much more complicated story, as well as a more interesting story.
So Saving Private Ryan gets 3 points for acting, cinematography, and the opening sequence, while Matrix gets 4 for style, story, complexity, and cinematography.
It pains me that I have to keep voting for it, but...
The Matrix.