The category of underlayment has a couple of subcategories. All kinds of synthetic underlayment are created from plastics to give them water proofing, but vary in thickness, slip resistance, and breath-ability.
Non-Breathable Synthetic Underlayment
A non-breathable synthetic underlayment has characteristics that resemble a tarp. This is advantageous as, like a tarp, it is completely waterproof. Unlike asphaltic underlayment, water saturation will never pass through a synthetic underlayment's barrier unless it makes its way in through the seams (each roll is overlapped with the previous but still represents a seam). The only downside of this type of underlayment is that the roof must "breathe" through the seams and roof ventilation. Trapping heat in your roof is always bad for your roof, and for that reason, a non-breathable synthetic underlayment is not the perfect underlayment. However, it is orders of magnitude better than asphaltic felt. This is the recommended solution for all steep slope roofing assemblies where you are planning on occupying the building for 5-10 years.
Breathable Synthetic Underlayment
This is the best category of synthetic underlayment. Despite the name of this type of product, it is completely water proof. The difference is that it features technology which prevents moisture from getting in, but allows air to escape. This allows the roof to "breathe". The most important part of this feature is that hot air can easily escape the roof which keeps the roof's integrity in tact long term. This is a recommended product for your forever home/building.