The only potential differences I can see in this would be in gimbal range and the ability to turn off certain engines. The falcon 9 uses 9 individual engines which allows it to turn off as many as it needs to in flight to either land, re-enter the atmosphere or in case one engine breaks. A multiple chamber single engine could do this with some modification however at the point where you are putting separate on the fuel lines it would essentially become individual engines. For gimballing purposes, using one pump system would only really work if all of the nozzles are connected and gimbal together. Again; if they were to gimbal separately the pump system would likely need modification which would essentially result in individual engines.
The only real advantage of using a single pump system is that it would ensure a stable fuel flow to each chamber. This is brought up in the RD-170 Wikipedia page "Several Soviet and Russian rocket engines use the approach of clustering small combustion chambers around a single turbine and pump. During the early 1950s, many Soviet engine designers, including Valentin P. Glushko, faced problems of combustion instability while designing bigger thrust chambers. At that time they solved the problem by using a cluster of smaller thrust chambers." (Wikipedia, 2017). Due to fuel flow requirements of large thrust chambers, fuel flow may happen unevenly between the separate chambers if not enough fuel/uneven amounts of fuel are pumped in. The use of a single pump ensures that all chambers receive an equal amount of fuel pressure.
Using a single large pump system over multiple chambers also likely helps keep the cost and complication down in the rd-170. This design however is not widely used today in currently planned and used launch systems. The only system currently using something like this to my knowledge (please correct me if I'm wrong) is the RD-180 used on the Atlas launch vehicle (https://en.wikipedia.org/wiki/File:Rd180schematic.png).