The goal, both in football and in hockey, is one - to score a goal and not allow you to score a goal. So much for all the similarities.
Here's what's different. Yes, almost everything: grass and ice, legs and hockey sticks, boots and skates, equipment, rules, atmosphere at matches, passion and heat. These are completely different games. And different people play them. In hockey, players are bigger men than football players, but there is more drama in football, from my subjective standpoint. Hockey is cozy and intimate. The fanatier is sluggish or walks a little, because he feels a special aura here, understands the rules of the game, not on the ice, but in general. And on football SUCH scale, SUCH fans, SUCH passions. No hockey match will bring our 800,000 people onto the streets of Moscow (and millions throughout the country!) Without any puppeteers and movements, after winning the quarterfinals of the 2008 European Championship over Holland! And this is not an isolated case. When, after the first Russian UEFA Army Cup in 2005, hundreds of Russians took to the streets. And this is only in Russia, where football is not very developed.
Football, one might say, "broke away" from rugby. So you can dig deeper - and try to compare hockey and rugby (although I don't quite understand why :)).
There is one similarity - both team games. Everything else is different. Although I am not a sports expert or even a fan of these sports. Basketball is somehow closer, maybe because I myself played for many years.