Different running shoes are designed to do different things. I'm a high mileage runner with high arches and, thanks to my orthotics, a neutral gate that likes highly cushioned shoes. For me that means Nike Air Pegasus or a brand and shoe that provides the same features, high arches, neutral gate & cushioned.
You need to understand what features your adidas brevard's have that you like and if the Brooks have the same features you'll probably be OK. I have also run in Brooks shoes and have liked them very much. Several years ago Brooks went more to motion control and stability shoes which just don't work for me so I switched to Nike. I also run in New Balance. I suggest you check out www.roadrunnersports.com as they do a nice job of grouping and comparing different shoes by their features.
Good luck.