I've had both of those tires when I use to live in the Desert of California due to Goatheads:
Tribulus terrestris - Wikipedia, the free encyclopedia These suckers penetrated the Conti Gatorskin like nothing, in addition the Gatorskins had fragile sidewalls like you have on your current tires. The Armadillo withstood the goatheads without ever having a flat in over 15,000 miles of using them, the tires last about 5,000 miles. And the Aramidillo has a very tough sidewall. Just go to an LBS that sells both and feel the difference in the sidewalls between the two. BUT, the tough sidewall makes for a very harsh riding tire and difficult to put on, plus the Armadillo is very heavy, about twice as heavy as a Gatorskin. But you have to weigh out if you want the leading tech for flat protection or a lighter tire that's decent protection, but not the best.
There is another tire you may want to consider called the Kenda Konstrictor. This tire weighs as much as a Conti but has the stronger sidewall...again though making the tire a pain to put on, but worth it when you don't get flats. The Kenda is way tougher then the Conti and will last longer.
Either the Specialized or the Kenda will be easer to install if you get one of these VAR tire tools:
randonneurextra: Making life easier: VAR tire lever