This debate rages among roofers who have preferences, so you might say that both types of underlayments have their own fan clubs.
Synthetics are favored for myriad reasons: they install fast, remain steadfast in high winds and when exposed to bad weather, they offer better traction that asphalt-saturated products. But synthetics come at a cost: They’re more expensive, especially if you choose a synthetic that is categorized as a vapor-permeable product.
Asphalt felt, on the other hand, has an older track record, it's also easy to install and you won't go broke if you choose this material. Asphalt-saturated layment was the only option on the market until around 15 years ago when synthetics--known also as non-bitumen synthetic underlayment--became available.