Hi Tim,
The Earth's atmosphere is more than just a protective layer, it's what we breathe. So, if the Earth were suddenly to lose its whole atmosphere, none of the plant or animal life on the Earth would survive, including humans. The Moon doesn't have an atmosphere, so the Earth would end up being more like the Moon if it lost its atmosphere. Without an atmosphere, the Earth would be completely devoid of any plant or animal life, just like the Moon. In additon to allowing us to breathe, the Earth's atmosphere causes most meteoroids become meteors and burn up before they ever hit the ground, so without an atmosphere, eventually the Earth would become covered in impact craters, just like the Moon, or the planet Mercury.
Kris