When it comes right down to this question, there is not that much difference between the two different varieties of roulette, or any of the other variations of the game. The main difference is in either how bets are placed, or how the wheel is set up. The rest of the game itself tends to work in much the same way.
The table is divided up into sections, each one with a number in that represents a section of the wheel. Players put their chips onto the table in the section they think will be selected and all bets are in, and a ball is dropped into the wheel. The ball stops on a particular number and any bets that are in that space on the table are declared wins. The differences come with how many sections are on the table and how many sections are on the wheel.
In some variations there are multiple ways that the players can bet. They can put a bet on a number coming up between 1 and 10, for example. They can choose whether they think it will be red or black that is selected or they can place ‘corner’ bets if they think there is a chance that one of two or four numbers will come up. In European roulette, the wheel has an extra space on it, the zero, which means that everything on the table is lost and play begins again. On an American wheel there are two such sections, the zero and the double zero. So the odds are much better if you are playing European roulette.
Whether you're playing in a casino or on your phone, it's always best to check the type of game before you commit any chips, just to be sure that you are happy with the rules before you begin.