In short, no. In the domain that we're talking about (website optimisation) bandit algorithms and A/B testing address exactly the same problem, and bandit algorithms do so more efficiently.

Bandit algorithms actually arose out of hypothesis testing when people started to think about stopping experiments early. One of the first papers on this is Chernoff's "Sequential Design of Experiments" (http://www.jstor.org/pss/2237415). Reading just the abstract on JStor will give the flavour of the idea, and bandit algorithms are the natural extension of this of this idea to a continuously running problem.