A "cause" means more than a reason--- it's like a purpose. Or it can mean the reason something starts. "What's the cause of all this trouble?" Means "What started all this trouble?" But in the phrase "lost cause," it's referring to a purpose. For example, stopping global warming could be a "cause." Lot's of people would say that you can't stop global warming, so they'd say it's a "lost cause." A "lost cause" is a purpose that you can't accomplish.

Yes,as a noun,cause means same as reason,but it can be used as a verb also - to cause, and as a conjunction, shorter version of because.