Corps style means that there is a lineage traced back to military drum and bugle corps that promoted musical excellence and marching precision. That's morphed over the years with the influence of college marching bands, music education programs, modern dance, Winter Guard International (WGI), and Drum Corps International (DCI). You may have read in the paper where two of the Marching Patriots performed in the Open Class division of DCI this year.
Here's an example from this season's DCI World Class champion's Finals encore (keep in mind that these are kids less than 22 y/o...skip to 2:30 to see what I'm talking about in regards to musical excellence):
http://youtu.be/AJ8fn-wXCqY
Show bands have their roots in jazz tradition and southern brass bands...more for local events, celebrations, etc. It's more along the lines of entertainment rather than musical/marching excellence.
These groups are both musical/marching groups, but of totally different genres.