Einstein has two major theories: (1) special relativity, and (2) general relativity.

The special relativity reduces to Newton's three laws of motion, when the speed of the moving objects are far lower than the speed of light;

The general relativity reduces to Newton's gravity theory when the gravitational field is weak.

So I would rather not say that the physics laws deviate from Einstein's theory; more precisely, Einstein's theories are "always" correct (with a catch I will explain in the next paragraph), but they become the more well-known theories of Newton under the condition of low speed and weak gravitational field. In our daily life Newton's theories are good enough, because we indeed move much slower than the speed of light, and the gravitational field from the earth is much weaker than the threshold to observe an obvious relativistic effects.

There is indeed a catch for applying Einstein's theory, more precisely, the theory of general relativity. The reason is that the general relativity is incompatible with quantum mechanics, which is the theory that describes physics at very small length scale. Hence the general relativity in principle would fail at extremely small length scale (the so called Planck scale). But again this Planck scale is so small that in our daily life we do not need to worry about the failure of Einstein's theory.