You are correct that H-U =pV. This means that ΔH - ΔU = Δ(pV). For a phase change at constant pressure this means that:
ΔH - ΔU = pΔV
So, the volume of your solid changes as you transition from diamond to graphite. Under normal conditions, ΔV is small, so the difference in ΔH and ΔU is also small. However, for large pressure, this difference is no longer small (as you can see from the equation above).