Rugby: Big, very strong men. If you can't run for 80 minutes straight, you can't play good rugby. Little safety gear, 2x40 minute periods. Multi skilled, super-fit athletes. No head-high tackles or tackles when ball carrier is in the air. Ball gets recycled after tackles in hotly contested ruck / maul situations. Sustained periods of scrumming / rucking / mauling uses every muscle in body for long periods of play. Exhausting aerobically and anaerobically. And still a lot of big hits.
American Football: Huge, super-strong men. 4x15 minute periods, with lots of additional stoppages. Focus on bursts of power and speed. No sustained periods of play. Much harder hits (because of impact-dispersing amour) but footballers have nowhere near the endurance level of rugby players. Potentially huge damage done to players bodies in blindside tackles on any given play. Pads help absorb impact, but MASSIVE impacts are common. Fatalities would occur every match without helmets, neck supports and pads.
A friend of mine who played both sports summed it up very well. After a football game, you hurt where you expect to hurt. After a rugby game, you hurt EVERYWHERE.