I have both. While I admit that the Remington is the more "modern" of the two with its top strap, fixed rear sight groove, and easily-withdrawn cylinder base pin, I personally prefer the 1860.
The 1860 has better-feeling grips and more natural pointing ability.
The loading lever feels more substantial on the 1860, and I have seen several reports of the loading lever on the Remington breaking at the pivot screw.
The fact that one can press out the wedge and remove the barrel means that a ball that creeps forward under recoil, or is pushed partially into the barrel because of a squib or mis-fire, can be cleared by simply removing the barrel to free up the cylinder or to drive the ball from the barrel. In the same situation a Remington is most probably locked up until you can get it home to work on it with shop tools, and then you are forced to operate with charged cylinder chambers pointed at you.
Most importantly, the 1860 is more tolerant of fouling, with it's large diameter arbor vs. the Remington's thin base pin, and that the gap at the forcing cone can be "adjusted" by means of the wedge.
Most of the 1860 advantages are, as I see it, directly related to loose powder and ball. Were I to be buying a revolver with the intention of doing a cartridge conversion, then I think the Remington is a better choice. But for cap and ball I'll stay with my 1860s.