Asphalt is a waterproof petroleum product. Asphalt primer doesn't mix with water easily, and naturally wants to separate itself from the liquid. This effect is identical to other situations where you mix oil and water. Asphalt primer needs to bond properly with the surface of the roof to function. A wet roof is going to hinder and potentially prevent the primer from bonding to the roof. In addition, rain can cause the asphalt primer to start to wash away. Once the asphalt primer is dry, it's waterproof and water doesn't pose a significant hazard. However, you should only apply it on a dry roof on a day where rain is not likely.
The primer should form a smooth, flat surface over the entire roof. Creating an uneven surface can cause a few different issues. Some areas might be too thin, which would allow water to leak through the primer. Thicker areas might take longer to dry and may still be wet when the rest of the roof seems to be dry. This situation can allow the primer to move slightly when the roofing is applied on top. In addition, the uneven surface may make it difficult to apply the roofing materials properly.
The issue with the roofing may not be an issue with the asphalt primer at all. The roofing material must also be applied correctly for it to properly adhere to the primer. Similar to applying the primer itself, water can potentially disrupt the bonding between the roofing material and the primer. Asphalt primer is waterproof, but its surface can still get wet. Make sure the primer surface is dry before applying roofing material.
Visually, asphalt and coal tar look similar. But asphalt primer won't work with coal tar roofing. It's not an issue of whether the primer or roofing is properly applied. Chemically, asphalt and coal tar are incompatible and don't bond well. A coal tar roof won't bond to the asphalt primer, which defeats the purpose of using the primer.