Logic tells me God doesn't exist but my heart says otherwise.

[Chas]

What, then, is the significance of these experiences?

I think it probably varies from person to person but some indisputable benefits reported universally and throughout history include an enhanced sense of freedom or will power, determination in the face of fear or conflict, increased peace in times of mental distress, a broadening of one's perspective that can and has led to important discoveries about one's self or the world at large, and sometimes even the cure of physical ailments, psychosomatic or otherwise.

Those effects and their mechanisms are a fit subject for science. Each person's internal, irreproducible experience, however, is not.