Earlier quoted context omitted.
You are right about the equipment (even though obviously only concussion vs concussion + skull fracture is a step forward!), but the difference between football and rugby goes way beyond the equipment. In the former there's no lateral passing, which means that the ball carrier will always be a target for the whole defense. In rugby you can pass the ball, which makes the defense spread wider and doesn't allow them to…
Precisely. In rugby, it is much more stressed and emphasized to make secure tackles. In football, at any given point and time there is one person with virtually a huge bullseye on them and it is sometimes a valid tactic to lay down a more punishing hit (cause a fumble, make a statement, scare your opponent to make a mistake or drop the ball next time, etc) at the expensive of not making a "secure, form tackle" Ironic…
Football is a game of inches. There is huge incentive to slam someone as hard as you can to stop their forward progress and ultimately deny the first down. There's incentive for the ball carrier fight for the inches and "fall forward" (which makes your head more vulnerable). In rugby you don't care about fighting for the inches and hell, your teammates will shove you down once they're set up to ruck over you. Fighting for inches in rugby is a quick way to start a maul and nobody wants a maul.
Furthermore, football is played with greater momentum (mass * velocity). Those breaks between plays equates to six seconds of full blown sprinting followed by forty seconds of rest. In rugby you're running about constantly so you simply don't have the energy most of the time for those massive hits. As for the mass side of the equation, football players are on average bigger than ruggers.
People think rugby is so hardcore for playing without pads (and it still is) but the hardest hits I ever took were in football (and I played defense, for goodness sakes). This is the consensus between everyone I've met that's played both. You can play multiple rugby games in a weekend (happens all the time at fests) but more than one football game a week is unsustainable.