The relationship between the soliton dynamics provided by the classical sine-Gordon and massive Thirring models is exhibited. Solitons are characterized as classical relativistic particles through the consideration of their associated canonical realizations of the Poincaré group. It is shown that the soliton in the massive Thirring model determines two different kinds of relativistic particles from which sine-Gordon kinks and breathers may be reproduced. In particular, sine-Gordon breathers are characterized as composite systems of two solitons of the massive Thirring model. Soliton scattering in the sine-Gordon equation is described in terms of soliton scattering in the massive Thirring model.