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.

And as according to research, you can also see the various opinions given on what the two stands for and their respective examples in which constitute each category.