A batch reactor is like a giant washing machine. There is a big vat where all of the reagents are put and a big agitator that keeps them stirring. A batch reactor is great if a company wants to make small amounts of specialty chemicals one "batch" at a time, but not if they want to make the same thing over and over. This is because the reactor must be emptied and cleaned after every batch is made. This takes a lot of time and money, and every batch can be just a little bit different due to small changes in reaction conditions, equipment aging, or because the operator drops a little bit of something foreign into the reactor.
In a continuous process, the reactor is basically a long tube. The raw materials go in one end, react on their way through the tube, not stopping along the way, and the finished product comes out the other end. A continuous process works well, because it can easily make large amounts of a product with little attention from a careless factory worker, and the product usually tends to be of similar quality throughout the process. The down side is, like in the initial production of Butyl rubber, if the tube gets clogged, the whole system has to be shut down for cleaning, which can cost a lot of time and money.
From Frank A. Howard, Buna Rubber: The Birth of an Industry. New York: D. van Nostrand Company, Inc., 1947.