Not saying Earth masses less than the atmosphere.
Earth is in freefall, hence has no weight. You can place Earth on top of me, in which case it would weigh about 900 Newtons, still considerably less than the atmosphere collective weight. A 20 digit number of Newtons is more weight than zero.
On the other hand, weight is a force, no? Force is a vector, and adding all the force applied by the atmosphere pretty much (not completely) cancels itself out, leaving many fewer digits of actual net force (weight?) on Earth.