In algebraic geometry, a curve is called hyperelliptic if it has a 2-to-1 map to a line. Equivalently, a hyperelliptic curve is the solution set to a certain form of polynomial equation in two variables. In tropical geometry, however, these two ideas aren't equivalent: the second is strictly stronger than the first. Along the way to proving this, we will completely classify which hyperelliptic tropical curves can be embedded tropically in the plane.