I've heard similar expression given on difference between samatha and vipassana, whereas the concept is the idea of breath and reality is the sensory perception. I think that was mostly a misuse of both terms. Because samatha is calming or stilling as in the Patisambhiddamagga 'in the first jhana hindrances are stilled, in the second thought and evaluations' and the vipassana can be explained as many things but the practice of observing the arising, persisting and the cessation of thoughts, feelings and perceptions for that is called 'development of concentration leading to alertness & mindfulness (samadhi sutta). Therefore it is imo a misuse of both terms for whatever was called vipassana in the time of the Buddha is probably not what they called the development of concentration leading to mindfulness & alertness. There are much better explainations for these terms afaik.
As to how to resolve this in practice, i would use the training of 'vipassana' for mindfulness & alertness. I would do it while being percepient of the breath, ardent & mindful of thoughts, feelings & perceptions, their arising, persistence and cessation.
As to what do i mean by being 'percepient of the breath' i can answer that if i was training perception of Death i would be thinking and exploring that theme with intellect.
So anapanasati is in practice sort of mindfully exploring phenomena associated with the theme of breathing or the air element (sutta about Pukkusati for air element equated to in & out breaths, can't recall title).
Also one can be mindful of the difference between 'a thought' and classes and categorizations of thought. The classes and categorizations should be directed at the theme of breathing or wholesome perceptions like the body, impermanence, fading, cessation, on the lack of a self, etc etc, so one would think i am breathing.. making long or short breath or just think that 'there is breathing' or otherwise just be aware of sensations, arising, persisting and ceasing, this is what id think about.
If one notices hindrances like perception of attractiveness, lust, anger, restlessness or drowsiness, id advice directing to a different theme to establish perception of daylight or unattractiveness or even go study, recite texts, do walking meditation, look at stars and whatnot depending on what hindrance it is before proceeding.
Eventually i want the unwholesome themes to stop invading and as they get stilled pleasure will arise and one sits quite comfortably, mindful & alert.