1. The "invisible" bit is just pamphlet sales talk. They are indeed visible. Think of a thin rugby player's mouth guard, you would notice it. Ironically this draws more attention to your teeth than regular braces since people are curious as to what you're wearing whereas with regular braces it's clear in 2 seconds and you soon forget about them.
2. Self-discipline. I failed with them due to poor self-discipline of putting them in every day and replacing after meals. Of course this needn't apply to everyone but it is a problem for many and my orthodontist confirmed they have a high failure rate on account of this.
3. Duration - they take much longer and are not as versatile at making minor adjustments in the way one can with braces and inter arch elastics. I have tried multiple orthodontic systems unfortunately and can honestly recommend regular metal braces as the most effective and quickest system with reportedly best success rate.
Of those, Damon is arguably the best as it is fast and uses gentler forces. Invisalign and other related systems are somewhat "gimmicky"