I boil the main differences down to three basic items: Collar, Color, and Cut.
Collar - Probably the primary source of variety among shirt makers, the collar can also be the most crucial in accomplishing "the look" you are after. The more you compare the more you'll notice that the length, point, softness, and roll of a collar are often details unique to its maker.
Color - As you can see from the pictures below, color can vary slightly even within a unique hue. Of the four blue shirts I compared, not one is the same as another in its degree of colorfulness. Granted, this can change over time and exposure but not to the degree that these differ.
Cut - By this I simply refer to the patterned shape of the shirt as it relates to fit. Slim, full, traditional, athletic, baggy, and regular are all common adjectives when referring to cut.
Four distinct shade of blue from four different makers.
Classic LL Bean (not the disgusting non-iron variety). This shirt is baggy and soft and the lightest blue of the bunch.
Brooks Brothers traditional. A full cut with a perfect roll over a tie. This is the classic blue.
Close-up reveals improvement with age...fraying.
J. Press flap pocket. A slimmer fit than the Brooks but by no means too slim. Probably my favorite cut. Collar could stand to be softer and does not roll over tie like the Brooks. Most similar to the Brooks in color.
Ralph Lauren Yarmouth. Nice baggy fit without as much extra material as the Brooks. Collar is on the small side but still dresses up. The "greyest" blue of the bunch.
The most often worn of the above shirts is the Brooks with the J. Press coming in at a close second. All of these shirts are sized as dress shirts, which obviously helps in the fit area compared to some other brands that might otherwise be worth comparing. I've heard great things about the Hyde Park oxford from Lands End but I've never tried them. I do, however, have my sights set on a couple of Mercer & Sons OCBDs this year so I'm anxious to see where they fall in the mix. If you have a brand that you think is worth highlighting, please weigh in with a comment.