What is the difference between JBoss ESB Server (which is available here: http://jbossesb.jboss.org/downloads/)
and JBoss Fuse (http://www.jboss.org/products/fuse/download/)
How are both functionally different?
JBoss ESB is JBoss/RedHats old original ESB product , basically it's old and sooner they will stop the support.
JBoss Fuse is the latest one, RedHat would prefer to sell only this now. It's latest, it uses Apache camel for routing, which is smart and popular one, it uses OSGi which has a different deployment and packaging approach, at last it goes well with RedHats whole other set of offerings and generally it's a technology in trend.
In my terms, JBoss ESB is legacy ;) , JBoss Fuse is preferred and latest for integration solutions.