First of all, I guess with "classical physics" you mean "Newtonian physics". Of course, in Newtonian physics the special principle of relativity must also hold. In both Newtonian physics and special relativistic physics thus Newton's 1st Law is valid, i.e., there exists an "inertial frame of reference", in which a point mass moves with constant velocity, if it's not interacting with anything.
The difference comes with Einstein's additional postulate for special relativity, i.e., that the phase velocity of electromagnetic waves in a vacuum (in short "the speed of light") is independent of the relative motion between source and detector.
Together with the additional assumptions about the symmetries of space and time you find out that you either get the Galilei transformations between two inertial reference frames,
$$t'=t, \quad \vec{x}'=\vec{x}-\vec{v} t, \quad \vec{v}=\text{const}$$
or the Lorentz transformations (making the direction of the relative velocity that in the ##x##-direction),
$$c t'=\gamma (c t-\beta x), \quad \beta=v/c, \quad \gamma=1/\sqrt{1-\beta^2},$$
$$x' = \gamma (x-\beta c t).$$
The Galilei transformations of course belong to Newtonian and the Lorentz transformations to special relativistic physics, and in special relativity, the speed of light, ##c##, is a "limiting speed", i.e., nothing can move faster than the speed of light within an inertial frame of reference. There's no such limiting speed in Newtonian physics, of course.
