Let me give two examples, however not so elementary,

1. The proof of the degeneration of the Hodge-De Rham spectral sequence by Deligne and Illusie in positive characteristic.

The basic assumption in Deligne-Illusie's theorem is the existence of a lift over the Witt vectors $W_2$. Hard to state within varieties. Moreover, Raynaud gives applications to instances of the Kodaira vanishing in positive characteristic.

2. Grothendieck's study of the fundamental group.

The notion of an étale morphism can probably be discussed within varieties, but one needs formal schemes, and the invariance of the fundamental group by a nilpotent immersion. But at the end, the result can be stated within varieties!