How does the Buddhist outlook relate to the scientific worldview? 

A good question. 

In the following answer, I am working from my understanding of Theravada Buddhism, based upon the earliest teachings of the Buddha.

There has long been a tension and often antagonism  between science and monotheistic religion. Some might argue that Buddhism is not a religion, and on a definition of religion that equates it with belief in the supernatural and a creator god, then Buddhism must be regarded as a philosophy or way of life.  But this suggests that our understanding of what religion means has become much too narrow, because Buddhism certainly provides a comprehensive range of spiritual guidance and methods, and has many of the features commonly linked with religion.

There are religious groups who feel threatened by scientific models. One unfortunate current issue is that of Creationism (now also called intelligent design). To see a conflict between religious descriptions of the creation of the universe (and humanity) and scientific theories (especially evolution)  is to confuse a worldview based upon the primacy of religious belief  (often fundamentalist, simplistic and literal interpretations of scriptures) with scientific models based upon observation, evidence and reason.

Religious fundamentalism and scientific worldviews are not compatible for various reasons. Fundamentalist religious belief is fixed and dogmatic. Although scientific models can also lead to 'devoted followings', they are ultimately open to evidence which may falsify them. This is the principle of 'defeasibility' which gives science a healthy openness, and means that doubt and dissension are legitimate. Science proposes hypotheses and ways of empirically testing these, whilst religious dogmatism already knows all of the answers.

There is little or no conflict between science and Buddhist practice. Science is concerned with building testable models of  the way the observable world operates. Buddhist practice is about developing our specifically positive human qualities of wisdom, intelligence and skilful action.  Science strives to be objective and evidence driven. Buddhist practice develops qualities and insights which are subjective and not easily verifiable, except  in the way we subsequently behave and perhaps in a certain 'quality' of our presence.  Buddhism and science both share an openness to evidence, an empirical outlook, and an emphasis on practice rather than belief. Both Buddhism and science are based upon naturalistic worldviews and do not encourage superstition, notwithstanding that there are some propositions in Buddhism which do not fit with current scientific theory

Humans are born with what might be called spiritual aspiration. This is a natural part of being human and calls for specific guidance and practice. The aspiration to understand the human condition; to end suffering; to experience joy, peace and fulfillment and to develop wisdom and enlightenment, would probably not commonly be given as a description of the scientific enterprise. However, scientific endeavour rightly directed, may offer powerful insights into the material world and biological life, and is a worthwhile application of our powers of reason and observation.