It's really a matter of form and function, although there is always room to compromise and improvise. Kind of like why you are looking at getting a saucier instead of just using a sauce pan.
A 6-qt casserole has shorter sides and a larger diameter than a 6-qt stock pot/dutch oven/soup pot (depends on who makes it as to what it gets called - All-Clad calls it a casserole and a true casserole a "buffet casserole"). The difference between a 13" chicken fryer/saute pan and a casserole, everything else being equal, is really the handles. Since a casserole is "normally" cooked in an oven, the short handles allow it to be placed in the center of the oven rack for more even heating than a pan of equal size with a long handle - which will generall cause the pan to be in a back corner of the oven.
While the casserole is shaped like the saute pan (flat bottom and straight sides), the brasier is shaped like a fry pan/skillet (curved sides and smaller diameter bottom), and generally has a domed lid where the casserole has a relatively flat lid. Aside from the fact that a 6-qt brasier will accomadate "taller" food than a 6-qt casserole, the liquid/juices are concentrated in a smaller area - the same amount of liquid in the brasier will be deeper than in the same size casserole.
Do you need either one? It depends on what you cook, how often you would use it - and how much money you have to buy things you may only use "once in a blue moon".
"It ain't what you don't know that gets you in trouble. It's what you know for sure that just ain't so." - Mark Twain