At one level the relation between Science and Spirituality is very simple. Science is a study of natural phenomena, that is to say, what we can observe through the senses. It observes and studies phenomena through various tools available to it or devices new tools to augment its field of observation. Spirituality on the other hand is a search and a quest for Reality, meaning the Truth underlying all the various countless phenomena. In doing so it tries to understand that meaning of creation. Science studies the immediate and the intermediate causes of things whereas Spirituality studies the ultimate original cause of all creation itself. Science studies the field of observation. Spirituality studies the Observer himself. Thus seen the two complement each other rather than contradict.
However, the problem starts when the science in its search for the processes and the how behind every phenomenon stands at the doorsteps where it must either take a leap of understanding to know what lies beyond the field that it has probed so far or else remain contented with its limits and say that thus far can it go and no further. Spirituality tends to extend itself from the Source towards the least detail that has emerged from IT. Here again, it finds the threads of creation lost beyond a point and has to either make a leap of understanding or else say thus far and no further. This is how the two have so far tried to draw an invisible fence between their respective fields. Science takes for its field the objective world taking its stand upon studying only that which is observable and definable. Spirituality on the other hand accepts as its field the subjective field, that which cannot be measured. However, the common element in both is a search for Truth. In this process, they have both entered a field where they would meet one day.
It is the field of psychology where already they have begun to meet. Though they find the meeting ground slippery and uncomfortable, yet it is here in the isthmus of the mind that material science and spiritual science would meet or are already meeting. As to people they are in the habit of having opinions and views even about fields that they have not even casually glanced at. I have seen that very often people who have opinions about either science or spirituality know neither. Regardless of all this a new mind of light is beginning to arise and awaken in at least the elite of humanity who beautifully harmonise and synthesise these two apparently distinct fields.
In other words, everything in creation has a material side as well as a spiritual truth that upholds it. Connecting the two is a whole mass of subtle and occult forces that form a watershed area between science and spirituality. To take the example of the earth itself, it is at once a material and a spiritual reality. It is also an occult formation with a very specific purpose and function. I am not aware of any authentic books that describe the integral reality except Sri Aurobindo and the Mother, so would therefore strongly recommend reading especially all the writings of the Mother. When we read them, we see that the veil of ignorance is slowly lifted and the confusion of understanding is cleared and a new integral vision of things begins to automatically appear before the mind.