There's no real difference in meaning. "Successfully" is straightforward and it's the one you should use. "With flying colors" is a idiom. It refers to the days of sailing naval ships. In a battle, the loser was forced "strike its colors," take down its flag. The winner would return to port "with flying colors." "With flying colors" means not merely successful, but PROUDLY VICTORIOUS. It is usually a compliment. You don't say it yourself about OWN achievements, that would be bragging. Suppose an exam has a passing grade of 70. Suppose I get a grade of 80. I am "successful." A friend might say "I heard you passed, good for you." Suppose I get a grade of 100. I would never say "I passed it with flying colors." But if a friend might say "I heard that you passed with flying colors."