The armadillo is a tougher tire that's less likely to flat than the gatorskin, at the expense of ride quality. It's a rougher ride, caused by some of the additional stuff they do to give it flat protection.

The gatorskin is a decent anti-flat tire which rides like a regular tire. Same ride quality, pretty much the same rolling resistance, etc. This is the tire most people are happy with.

I'd only suggest the Armadillo to people who are already having serious problems with multiple flats, like the original poster, because of it's reduced ride quality.