The structure of the multiplicative groups of Z/pZ or of Z_p is the same for odd primes, but not for 2. Quadratic reciprocity has a uniform statement for odd primes, but an extra statement for 2. So in these examples characteristic 2 is a messy special case.

On the other hand, certain types of combinatorial questions can be reduced to linear algebra over F₂, and this relationship doesn't seem to generalize to other finite fields. So in this example characteristic 2 is a nice special case.

Is anything deep going on here? (I have a vague idea here about additive inverses and Fourier analysis over Z/2Z, but I'll wait to see what other people say.)