External monitor has "snow" at 1080p/i when connected to MacBook Air
Likely to be the adaptor cable if it's cheap - might not be able to handle the HD throughput (although outputting to HDMI, you'd think it'd be able to). 

While this sounds counter-intuitive, try moving around the alignment of the second screen as this might ditch the 'snow'. 

Also, have you tried the screen with any other HD inputs?