Starting from my own end, here is my definition and understanding between the two.
Casino Gambling involves Playing games that are base on luck factor and sometimes our skills also work out in this, it does not involve live event.
Sport Betting simply means, playing on live matches which involves you and a particular player or team against each other.
Well, first of all, let me mention that this is a topic that has been discussed on this board on several occasions before, in fact, I was actually one of those who have started a thread on exactly this same subject before, but all the same, the forum is an online platform which new people are joining almost every day, so, it's never bad to discuss what have been previously discussed as far as newbies learn from it.
Casino games are pure luck based games, you talked about skill in this area, but to be honest with you, I do not see any area of casino games where skill helps the player to win, winning and losing in this type of gambling games totally depends on how lucky and unlucky the player is.
While sports betting is more like a knowledge based game, this simply means that, the gambler's knowledge on sports teams and clubs, and every other thing about sports generally, helps the player to make better prediction and stand a better chances of predicting correctly and win his or her bet.
And as a correction, sports betting is not just about live betting, people can bet on sports matches that still have even up to a month before the game will be played, all that matters is that the player is cool have his or her money locked in a bet for that long.