Both sentences are very similar in that they both convey the idea that a Disease is spreading quickly. The better sentence to use to convey that idea is the second sentence "The disease is spreading very quickly." The first sentence "The disease is being spread very quickly" gets the same idea across, but when "..being spread.." is used it seems as if something or someone is actively helping to spread the disease.

"spread" can take an object or not. The first sentence is passive. There is an implied agent - the disease is being spread by someone who or something which is not mentioned. The second sentence is active without an object. The disease is moving through the body.