“Roman Church” refers to the Church of Rome, that particular or local church of the city of Rome (the diocese of Rome). This particular Church is the head or chief of all the particular Churches.
“The Church” is ambiguous in the sense that it can refer to a particular church or the universal Church, although nowadays it is usually used in the latter sense.
The universal Church is sometimes referred to as the Roman Catholic Church, due to the headship of the Church of Rome and the communion of the rest of the Church with it, but technically it only refers to those particular Churches which operate under the laws and customs of the Roman Rite. There are other [adjective] Catholic Churches with their own customs and rites [Ukrainian, Syro-Malabar, Melkite, Maronite, etc.), which are all part of the one Catholic Church founded by Christ.
But yeah, one shouldn’t take Roman as an insult if you are a Roman Catholic and if you’re, say, a Maronite Catholic, it is probably just a matter of ignorance, not malice. On the other hand, when someone says “Romish” nowadays then they are usually using it as a pejorative.