Not a complete answer, feel free to edit. (Likely, the answer is not known anyway.) The distance is at least 2. Look at both spaces as $C(K)$ spaces. Corresponding $K$'s are not homeomorphic, see the discussion here https://math.stackexchange.com/questions/207435/isometry-between-l-infty-and-ell-infty
So by Amir-Cambern theorem (near isometry property of $C(K)$ spaces) the distance is at least 2. The theorem says if there is an isomorphism between $C(K_1)$ and $C(K_2)$ with distortion strictly less than 2, then $K_1$ and $K_2$ are homeomorphic.
