An Elbrus climb to both of its summits is usually preferred by people who strive to accomplish all their endeavors in order to eliminate any understatement, or by people eager to experience as much of the world as possible.
Indeed, East and West summits of Elbrus differ substantially, and both of them offer quite different views from their tops. However, this is not a reason to overestimate oneself: standing on one of the summits’ top rare climbers can find psychological and physical recourses to venture for the other summit right away. Most climbers think only about getting back to the refuge as soon as possible to get some hot soup and internet.
Anyway, it makes more sense to spend another 2–3 hours to climb the other summit right away, from the Saddle (with an altitude gain of as little as 300 m) than from the refuge at the distance of 2000m. This way you will avoid the feeling of understated accomplishment.