Depth matters for two reasons: frost protection and load protection. A drain laid too shallow under a driveway will be crushed by vehicle weight over time, which is one of the most common causes of collapse we find on properties where a garden was later paved for parking.
Where a shallow run is unavoidable — coming out of a low-level extension, for instance — the pipe should be surrounded in concrete or protected with a reinforced concrete slab bridging it, so wheel loads are carried around the pipe rather than through it.
Depth also drives the repair options. A shallow defect is cheap to excavate; a deep run under a conservatory is exactly where no-dig lining pays for itself many times over.