The advantage is that if you go off road like many of us do/did, not having a flat tire is a big benifit. I had them all foamed in my Imp, and figured it was well worth it!
Downside is that it does add weight, but that verses having a flat off road is pretty slim. As far as ride, when you have the tires filled, you tell them what pressure you want them at, and that will affect the ride as well. The tires still flex and with a lower pressure they flex more verses higher pressure because the foam sets up at that pressure. In my case, at sub zero temps, the ride is pretty much like a brick with or without foam when the tires are cold and hard.
Can't remember the company in Fairbanks, use to be Kelly Tire accross from Gene's Chrysler, I think they are Bridgestone now... They foamed my last bunch of tires and did a good job and the price was inline with other shops that I ran into..