The fact that you compare working for and established company to "curing cancer" and going to work for a start-up as "developing a sexting app" shows little knowledge of what start-up and established companies are actually doing. The fact of the matter is, working for a larger established company usually consists of maintaining or making trivial enhancements to existing software with the occasional new product being developed. Working for a start-up, however, usually includes a rampant amount of innovation simply because start-ups don't have much money to advertise their new products. The result result of this is they have a need to develop more interesting and innovative products in order to be able to compete with established companies.
Another thing worth mentioning is the diversity that start-ups usually have, need I remind you that Tesla motors was a start-up, and many of the technologies, including some which show promise of curing cancer, were also developed at start-ups.