Looking around numerous sources, they all seem to answer this question the same way. Generally the Sith Lords are people who study the sith philosophy. So there isn't really a difference between "Sith" and "Sith Lord" unless you're talking about the actual race (which is EU canon):
The Sith are dedicated to sith philosophy and to mastering the dark side of the Force. The Sith members, known as Sith Lords or Dark Lords of the Sith, traditionally use the title Darth-prefix before their Sith name. Sith - Wikipedia
Sith tend to be a bit egotistic, and generally "up their own bottoms", to the point where numerous times the apprentice Sith Lord gets tired of his Master and wants it all to himself. So it doesn't really surprise me that they would call themselves "Lords" just to make their presence more important.
After the "Rule of Two", I can't seem to find any instances of someone following the ways of the Sith without the prefix "Lord", although the more studious people on this site may be kind enough to change this point if there is.
It seems that subsequent to Episode VI, the Sith ceased to exist, and new dark orders were created (or possibly just came to the fore), such as the Knights of Ren.
Kylo Ren's agenda seems to somewhat differ from Snoke's so maybe his plan is to rejuvenate the Sith order and "carry on what his grandfather started". Who knows, this is just my speculation.