You could say that Buddhism provides methods to use life as a lesson, every moment is a learning opportunity. That's why the texts and talks are called 'teachings' - by learning to change our perspective on life, we can deal with it more skilfully.
Does Buddhism recommend itself as the best approach to reach enlightenment, or can this fulfillment be achieved outside of Buddhist schooling?
Buddhism contains teachings that are singularly designed to help human beings get out of this endless wheel of suffering (in Buddhism called Samsara). By learning to see things from the perspective that There is suffering, Suffering is caused by craving/clinging, There's a way out of suffering and The path to end suffering is detailed [in eight 'components'] we can skilfully reduce our suffering/anxiety/dissatisfactoriness of our lives.
Most importantly, the Buddhist teachings were not designed to be believed in. The were designed to be used. So we can choose to learn the doctrine like any other religion, and believe it or not, or we can put the teachings to practice. Only then will they have an impact on your life.
Part of Buddhism ("being a Buddhist"), involves taking on five (or more) precepts, similar to other religions. These include refraining from killing, stealing, sexual misconduct, lying, and taking intoxicants. However, to practically train the mind, meditation is an essential component of the Buddhist training. There are thousands of techniques that can be put to use, most helpfully aided by an experienced teacher.
What is different about being a Buddhist than an atheist?
The question does not entirely apply, but suffice to say that atheists reject that there is any god that "rules the world" - Buddhist teachings include the existence of deities, heavens and hells (all plural), but that none of them "rule the world", that there is no one entity that decides over this existence. Important is to note that these deities are not worshipped as if they are above humans, and Buddhists have compassion for those in the hell realms.
NOTE: This is a very simplified answer in response to your questions. In summarizing this way, inaccuracies may have crept in, unintentionally. I encourage hearing more from a Buddhist, in person (ideally a monastic), to get a better understanding. (opposed to just reading about Buddhism)