So I googled A/B testing vs Multi-Armed Bandit, and ran into an article that's a useful and informative response to the OP: https://vwo.com/blog/multi-armed-bandit-algorithm/

edit: Ah, 'ReadingInBed beat me to it. tl;dr: Bandit approaches might not _always_ be the best, and they tend to take longer to reach a statistically significant result.