Shimano Nexus SG C6000-8R and SL-S503
This answer depends heavily on Shimano documentation and has a number of links to the Shimano website, which will inevitably suffer from rot. My apologies to all those reading this years from now when Shimano has changed all the URLs.



I think that in general the last couple of digits in Shimano component names/codes denote variations in the spec or quality level of the components. 

One can dig around in Shimano's tech documents looking for answers. Current URL is http://si.shimano.com/#/

The Line Up Chart offers some insight into the compatibility of current parts (note there is an archive also).

The Nexus Inter 8 Chart indicates that the SG-C6001-8R hub can be used with the SL-S503 lever.

I note that the the 8R postfix in the hub name denotes 'roller brake spec', but I'm not sure if that means it has a roller brake built in or it is compatible with a roller brake. There seems to be a specific compatibility with brake levers for roller brakes.

Update: after reading the Sheldon Brown / Harris Cyclery page on roller brakes, I found out that the 'R' postfix hubs are compatible with an add on brake unit.