Yes, both are equal for small angle changes. Exceptions do apply, especially in transsonic flow.
Both changing the stabilizer incidence and the elevator deflection will change the lift distribution between wing and tail surfaces and will trim the aircraft for a different angle of attack. Changing the incidence gives the empennage a new angle of attack while changing the deflection will change the camber (and with it the zero-lift angle of attack) and the angle of attack of the empennage. For small deflections the effect is the same; however, the change in stabilizer incidence is smaller than the elevator deflection change for the same effect in proportion to the square root of the relative elevator chord.
Say you change the stabilizer nose-down rsp. the elevator trailing edge up. The tail will produce less lift / more downforce and the aircraft pitches up. Now the angle of attack of both the wing and the tail will increase which increases lift on both until the distribution of lift between both is as before - after all, we did not change the position of the center of gravity, and the angle of attack will change until the center of pressure is exactly below or above the center of gravity again. There the aircraft finds a new equilibrium speed and will fly on in its new attitude.
Nevertheless, depending on the initial trim point, the aircraft will now climb or sink because the thrust does not match the new drag at the new trim point. The aircraft will settle at a new vertical speed, too.
Before you think that one of them would be enough: Both are needed because each is a specialist for tasks that the other would handle poorly:
- The elevator moves quickly and needs small actuation forces, so it is built for manoeuvring or gust response.
- The stabilizer moves slowly but can effect much higher lift changes. It is meant for retrimming during or after changes in configuration. Deflection of the lift devices on the wing will shift the center of pressure backwards, so in order to balance all lift forces, the tail now needs to produce massive downforce, something only the adjustable stabilizer is capable of.
This answer covers these aspects in more detail.