Deck Armor breathes and allows moisture to escape from underneath, while blocking water from above. Felt does not. However, deck armor considerably more expensive. I use it whenever possible.
How's the moisture get underneath if not through the material?
I have only used synthetics a couple times and I was under the impression it was suppose to be a water tight material, all tho I have never seen it in action because it did not rain on either roof that we used it on until after the shingles were installed.
Felt holds water out 'stops moisture from getting underneath' when properly installed and #15 felt will out last a 30 year shingle and a #30 felt will out last a 50 year shingle.
So unless your talking Tile or Slate that will set for a hundred years if properly installed and maintained then the regular old time tested and proven to work #15 and #30 felts are all you need.
Ofcourse in situations like mentioned by Dale in which the underlayment is going to be left exposed to the weather for a long period of time it would definately be worth the extra cost to use synthetics.