Corporations have management bottlenecks; often times it's hard to work up because the positions are filled and the people in them aren't leaving. In a startup environment, if it's small, you have a chance to dig in and help the company grow. You also have the opportunity to build relationships with the executive team and show them you are worthy of being one of them. Now a days, companies aren't promoting within as well and are looking to outside candidates with experience, this can really hurt your career and keep you stuck.
Startups usually ( and I emphasis usually as sometimes a negative founder can taint the culture from the beginning) have a positive, fresh and exciting energy to them. An established corporation's energy is most often stale, unhappy and stuck as it is a reflection of the majority of the employees. Everyone walks around with a fake smile and it's really toxic to its core. Trust me, you don't want to absorb those energies and taint your own personal outlook. If a company's culture is toxic, run.
And this is my favorite, there is NO Human Resources department in early startups which means you get to interview with the leaders of the company. This interview process is often more organic and less robotic. The leaders are more likely to be visionaries and take risks, this means they can see potential and are willing to bet on someone they really like. HR people have not been trained to think like this. And most importantly, you get a chance to connect with the leaders and they you. Connection is just as important as skills. Nothing kills a team faster than lack of connection. Also, if there is no ATS, you're much more likely to land an interview, AND the hiring process will be quicker.
Sure, there are bigger risks with a startup. Many startups go under for various reasons so do your due diligence when researching the business, its financial backers and managing team.
However, keep in mind that there is no security in this world. All is subject to birth and death, including large companies.
If you're job hunting and want to find a startup, check out Angellist.com and best of luck in your hunt!