That oppresses, like the Heft Of Cathedral Tunes— (3-4)
Those tunes have a way of making the speaker feel oppressed and burdened by the heaviness of their sound. So, while everyone else is presumably going off to church services in response to those tunes, the speaker is left feeling isolated by those same sounds.
Heavenly Hurt, it gives us—We can find no scar, (5-6)
Since the light gives a "Heavenly Hurt," we understand that the speaker is feeling this sort of hurt in a more internalized and therefore more isolated way. All of the evidence of this hurt is inside since she can "find no scar."
But internal difference—Where the Meanings, are— (7-8)
The "internal difference" tells us that the speaker senses this difference in an isolated way that can only be understood by her. And since it's where the "Meanings" are, we also understand that her perception is part of this "internal difference" that's inside of her.