Established mobile network operator and MVNO, what is the difference in terms of communications quality?

There is no difference. 

Indeed, as they do not operate any technical infrastructure of their own, MVNOs - virtual mobile network operators - use the network of incumbent operators. Connection and throughput speeds therefore remain exactly the same.

Differences are about price, quality of service and 4G network coverage. 

Some operators without infrastructure are specialised in providing packages intended for companies (“business MVNOs”) and use an aggressive pricing policy for a level of service just as interesting as that of the incumbent operators. However, faced with competition from these new players, operators are increasingly trying to match prices.

Note that free mobile applications now make it possible to measure and compare the network coverage of the different mobile network operators, established and MVNOs alike.