First off, let's try to stay on subject, folks. The Question was the difference between the 1851 and 1860, those are both Colts. dallas.moore was not asking about Remingtons, and which is best is another issue entirely.
The 1851 Colt Navy was called a Navy because of the US Naval engagement with the Mexican Navy scene on the cylinder. This had little or nothing to do with whether or not the US Navy purchased or preferred that revolver, and the Navy issued single shot pistols for boarding parties, even after changing to cartridge, they issued Remington Rolling Blocks.
All Colt Navies were 36 caliber, except, as another mentioned, some factory experiments. The 1851 had the octagonal barrel and hinged loading lever. The 1861 Navy had the round streamlined barrel with a creeping loading lever similar to the 1860 Army.
The 1860 Army, despite having the Naval engagement scene rolled onto its cylinder, was not called a Navy. An Army, no matter what company or design, was defined as a 44 caliber revolver with an 8 inch barrel. Unlike the later Government purchases, where different vendor's guns had to interchange parts for an identical design, the Civil War era revolvers needed only meet the barrel length and caliber spec.
A "Navy", named because of the Colt cylinder engraving, was defined as a 36 caliber revolver with a 7-1/2 inch barrel, no matter who made it.
Notably, Pietta is marketing a 44 caliber "Navy". It mounts an Army cylinder and frame with an octagonal 44 bored barrel of 1851 style, and a Navy sized gripframe. It is not something Colt ever produced, nor even the CSA.
Most CSA revolvers were copies of the Colt Navy in 36 caliber or the Dragoon as a 44.
The action parts of a Colt Army and Colt Navy are identical except for the rebated cylinder and notched frame. A Navy barrel and cylinder can be mounted on an Army frame and funcion perfectly. Not vice-versa, as the Navy frame will not accept the Army cylinder. This is true of the Italian reproductions, but might not have been true of the original Colt, unless the barrel pins were in the same location. I no longer own originals of both to compare.
For that matter, the action of the SAA is identical except for the hammer, hand and spring and hammer pivot screw. The mainspring, bolt/trigger spring and screw, trigger and bolt, trigger guard and backstrap and wood grips are interchangable between the 1851, 1860, 1861, 1871, 1872 and 1873 SAA and the conversions.