Hello Veritatis cupitor!
I don't think that science and mysticism is in conflict, but I do think they are incompatible. Note that I am refering to the real stuff here, not the phony, silly mysticism such as "quantum mysticism" or "UFOlogy" or "Astrology" or something - because they clearly conflict with science. To the extent that mysticism is a purely mental experience, absent of any formation of concepts about physical reality, then it does not conflict with science.
Much of the problem with mysticism however, is that mystics tend to feel the need to interpret their experiences in meditation/hallucinogenic trips/trances, and generally do so within the framework of their individual religions, cultural traditions and personal beliefs and preferences. As a result disagreements occurs, even within all major religions, as well as disagreements with the scientific view of the world. Many mystics believe that their experiences give a knowledge of fundamental reality, they believe that the reality they experience during these mystical experiences is transcendent and the source of all existence. Such supernatural beliefs with no other basis other than personal impressions gained during these experiences are widespread.
Also, the notion mystics tend to have is that reality is a structural hierarchy with consciousness/spirit at the bottom, from which all else emerge. This mystical hierarchy is incompatible with what science infers from its data, which goes like this; matter - life - consciousness, with no spirit and perhaps no separate entity we can associate with consciousness. So, without any evidence or other rational basis, science cannot accept the notion of transcendence. In fact, much evidence indicates that the mystical experience is all happening inside the head.