The Purple Cassock is called the "Choir Cassock". It is used when a bishop is saying Mass or in any ceremony when he would be seated in Choir. It is sometimes worn after and before these ceremonies when it is inconvenient to change.
The Black Cassock with Color Trimmings (and sometimes a shoulder cape) is called the "House Cassock" and is worn throughout the day when a bishop is not participating in any liturgical ceremony, but is performing other duties. Sometimes bishops also use a plain black cassock as their house cassock. The trimmings and the sash color correspond to their office.
Minor Prelates (Monsignori who are not bishops) have the privilege of using some Pontificals, including the use of the cassocks.
For example, if Bishop Williamson were assisting in choir at an ordination (but not doing the ordaining), he would wear the purple cassock, rochet, and mantelletta along with a purple biretta. If he were the ordaining bishop the purple cassock would go under all of the rest of his Mass vestments.
Let us say an old priest was given the title Monsignor by the Pope. He also could wear the above accouterments, except he could not wear a purple biretta (only a black biretta with a amaranth red tuft. If he were a Pronotory Apostolic de numero (a Curia member with duties in Rome) he could wear the rochet and manteletta. If he were a supernumerary (honorary title outside of Rome) he would be restricted to the surplice.