There are at least two major similarities between the characteristics needed for these two.
First, both of them need to have a great deal of self-confidence. Entrepreneurs need self-confidence because many new businesses fail. They need to be able to understand that they could fail and push ahead even so. Similarly, professional athletes know that the vast majority of people who try to make it in that profession fail. They have to persevere, just believing that they will be the one to beat the odds.
Second, they have to be disciplined. Entrepreneurs and athletes both need to have a great deal of focus so that they can do all of the small things that are needed to be successful. Athletes have to practice over and over and they need to do very hard work for conditioning. Entrepreneurs need to be disciplined enough to do all the hard work that goes with a business, keeping the books straight and other such unglamorous things. There is no one to force either group to stay and do these things, but they are necessary to succeed. Therefore, it is the disciplined people who succeed.
Most of all, entrepreneurs and professional athletes both need to possess a great amount of drive. Since both professions require personal initiative, neither entrepreneurs nor professional athletes can rely upon a stable structure to provide steady support. Like athletes, entrepreneurs are entering a market that fluctuates according to performance. If you do not perform, you do not get to reap the rewards that accompany the profession. In contrast to a more traditional job, there is not a steady paycheck promised just for showing up. Both career paths are high-risk, high-reward. The good news is that entrepreneurs are typically rewarded for qualities like aggression, intelligence, patience, fortitude, and so on. The bad news is that sometimes, the amount of work you put in will not necessarily be represented by the tangible results. The same could be said for athletes. Sometimes, the competition is just too fierce, no matter how talented or disciplined you are. In both professions, you need a great deal of grit and perseverance to make it through times that are difficult because difficult times are a given in either profession.