Which works better, competition or cooperation? The answer, without equivocation, is both. In some situations, cooperation is dominant, whereas in others, competition is.
Nowadays, we are living in a competitive world in general, and a competitive society in particular. There are usually many people "out there" who are qualified for the particular job or position or grant that we are seeking, and if it is something we deeply want then we need to do our best to stand out--to look better. In short, we must compete. However, competition requires to be healthy. Healthy competition can be useful. Working hard at something and competing against others who are also working hard can inspire us to push ourselves farther than we otherwise might. "External motivation" is not necessarily a bad thing if it prompts us to excel.
On the other hand, we might think that we can do a job by ourselves, and later find ourselves over our heads; or we might indeed be able to do the job by ourselves, yet it would have been faster, and easier if we cooperate with others. Getting cooperation from others is like growing extra sets of hands to finish a job quicker, or getting more brain power to solve a problem. It also seems that cooperation has an impact on individuals working together in several key areas. Not only does it create a more fluid leadership, but also it allows everyone to participate actively without fear of censure. Moreover, cooperation has an impact on an individual's perception of the work environment. In cooperation, people work together in order to achieve their goals. By helping you I help myself. We sink or swim together. This is a win-win approach--everyone wins.
Competition and cooperation are two very powerful tools, and learning when it is beneficial to compete and when to cooperate is a key to succeed in any project or job.