A phone app lacks many features that compasses have which are essential to orientation. Among the main ones:
- A transparent casing that makes it possible to read the map through the compass and align the grid to the needle.
- Numeric scales and rulers that enable you to make measurements on the paper map
- (not on all models) Mirror with sight lines or reticles that enables you to sight landmarks and read the compass needle at the same time for more precise aiming
- (not on all models) Mechanism to adjust dial for magnetic declination
Another quite important thing to know is that scattered around the world are magnetic anomalies, mostly ferromagnetic deposits. Considering electronic compasses drift pretty quickly and they need to be self-calibrated regularly, there is a chance that you could calibrate your electronic compass with a local bias, which would make it misaligned even when you left that area. This would not happen with a regular compass. The anomaly would skew the compass but it would return to normal afterwards.
Some anomalies are quite limited in range, but I've experienced a couple while sailing in the Caribbean that would disrupt the pedestal compass for at least a mile.
But more importantly, a phone will not work if the battery is dead. A compass will always work unless you accidentally demagnetize the needle.