When we say that two things are "equivalent" in physics, we generally mean the following:
$A$ and $B$ are equivalent if one can be transformed into the other using operations that don't affect physical meaning.
For example, we know that a measurement of the distance between two points is equivalent to a measurement of the time it takes light to travel between two points. This is because light always travels at the speed $c$, so therefore for any time interval $t$, there is a corresponding distance $d=ct$. In this case, multiplying or dividing by the speed of light is an operation that doesn't change the physical meaning of the measurement in the framework of relativity (this is, in fact, part of the rationale for setting $c=1$ in relativity, to emphasize the equivalence of distance and time intervals).
You'll notice, in the above, that two things can be equivalent without being literally the same thing. Distance is not the same thing as time (for proof, see the impossibility of traveling backwards in time), but every distance interval can be converted into a time interval. Any two non-identical things can be contrasted, even if they are equivalent. For example, we could say, "distance intervals, which are always nonzero for spacelike-separated events, stand in contrast to time intervals, which can be zero for spacelike-separated events." So there's no contradiction in saying that two things are equivalent and can be contrasted.
In our particular case, the Schrodinger and Heisenberg pictures are equivalent because any statement made in the Schrodinger picture can be transformed into a unique statement in the Heisenberg picture, by executing what is effectively a change of basis (which doesn't change the physical meaning). They can be contrasted because they are not literally the same thing - in the Schrodinger picture, the time-dependence of the statement is carried by the wavefunctions, while in the Heisenberg picture, the time-dependence of the statement is carried by the operators.