A running shoe is for *running* and is built for that purpose. WHat does that mean? In general it a lot lighter and sometimes, but not always more cushiony.
I run about 10 miles a week. If you pick your feet up a thousand times, you'll notice the extra few ounces. If you really need a cross trainer for all those mountain jaunts, get it becuase it will give you far greater ankle support and usually a tougher sole.