What could be the difference between contracting a disease and catching a disease?

There isn't any big difference. However, it looks like there are some diseases you can both catch and contract and others you can only either catch or contract.

Contract means[with object] catch or develop (a disease or infectious agent): Three people contracted a killer virusCatch meanscontract (an illness) through infection or contagion: He served in Africa, where he caught malariaThe principal difference is that catch suggests a transmittable infection, while contract can refer to a wider variety of diseases, including those that are not contagious.

More.catch malariadevelop cataractscontract tuberculosissuffer a strokesuffer from vertigodevelop myopia