What is the conversion rate of penalties in football versus ice hockey?
I realise that it would be different between, and dependent on the league but any two major leagues of comparable level would be interesting, for example, the NHL versus Premier League or La Liga.
It's way higher in soccer than ice hockey. In soccer, a penalty kick is quite close to an automatic goal, averaging around 85% in the English Premier League (source: My Football Facts), whereas in ice hockey, a penalty shot has a conversion rate of around about 30% (source: official NHL stats, via this excellent answer).
However, I'm not sure this is really an interesting comparison as the criteria for awarding penalty shots are very different between the two sports, so them having very different conversion rates doesn't indicate anything is "wrong". For example, what would result in a penalty shot in ice hockey (fouling a player on a breakaway) would get you sent off in soccer, potentially a much more serious penalty, particularly as in most professional leagues that would result in the player being banned for at least one match as well.
