Lets take an example of how science and spirituality will deal with a situation and then proceed to understand the differences or similarities between them, if any.
Let us for our example study 'Hunger' as the subject. Science, tries to understand how hunger pangs develop, how our body and mind is able to know hunger, what physical changes happen within our stomach and what communication takes place through our nerves. That's science for us.
Whereas spirituality tells us to experience hunger once.
Science is useful in situations where a person who has experienced hunger is curious to know how the whole dynamics of hunger work. Where as spirituality is useful, when a person wants to understand what hunger is.
Science deals with intellectually understanding a particular subject, whereas spirituality deals with the experiencing.
Science focuses on 'doing', spirituality focuses on 'being'.